Is it possible to remotely connect to the linksys device
Yes.
Most routers have an online administrator page that can be accessed externally by visiting your public IP (go check WhatIsMyIP.com too see yours).
It's even easier internally by accessing the internal IP of your router, which is in most cases either 192.168.0.1 or 192.168.1.1. If these don't work, you can find this IP address by opening a Command Prompt by opening start and typing cmd (Win Vista/7/8) or by pressing Windows + R and entering cmd. Once the command prompt is open, enter ipconfig and look for the Default Gateway address under Wireless LAN if you're connected via the WiFi or under Ethernet adapter for LAN connection if you're connected via wire.
If you browse to the address of your router, you shall be greeted by the Lynksis administrative page and you'll have to login. Read the manual or Google lynksis [your model, see box] default login to find out the password.

and see my online activity
This depends. Some routers allow to log the activity of the connected devices. Check the manual or the administrative page of your router for settings like this.
If I were to stalk you, however, I'd connect to your network and run a sniffer. A sniffer snifs all network traffic on the network you're connected on and most of them allow to filter on interesting protocols like HTTP (normal browsing network).
If he knows the password of your WiFi, he can connect to it and sniff your network. Sniffers are tools that listen to all network passing by, even if it's not being sent to you. WiFi is easy to sniff, as wireless data is sent in all directions.

and should I be concerned?
Do you trust this person?
You allowed him in your house and you allowed him to install the device. This means that there is some basic trust between the two of you.
Is there any reason not to trust this person?

What can be gained out of doing something like this?
The neighbour might be connected to your WiFi to steal your bandwith, i.e. he downloads movies via your network so he doesn't have to pay for it. He might also use your network for stuff he wouldn't like to see linked to his own network (e.g. criminal activities or surfing to adult websites).
If he's able to sniff your network, he could capture password on weakly protected websites (they use normal HTTP instead of HTTPS, which encrypts network traffic). If you suddenly were to see notification of HTTPS-sites being non-trustworthy anymore, although they have always been (e.g. Facebook), then he's trying to sniff the encrypted connection between those websites. If you use other services, like FTP to connect to a fileserver or RDP to connect to a virtual work environment, he could steal those passwords.
By listening on someone's network traffic, it's also possible to collect information about a person's behaviour. Do you check a lot of travel websites, adult websites, cat videos, ... This information could be used to make a profile of you. The advantage of this depends on what it would be used for.

If this issue is possible, how can I defend myself?
If you decided that you don't trust this person, there's a lot you can do yourself.
First of all, undo all possible settings your neighbour made, this means resetting the device.
Unplug all network connections to the device and reset it by sticking a pencil in the hole at the back where it says 'reset'. Hold this for 5-10 seconds. Most devices will notify you of the reset, e.g., with some blinking lights.
Follow the instruction manual of your device to reinstall it. In most cases you'll have to connect a device via cable or via a default WiFi and browse to the IP-address I mentioned before.
You'll be able to change the administrative password, name of the WiFi and password of the WiFi. Do all of this. Be sure to use a strong password (8+ characters, capital and small letters and a few numbers, maybe a - or _). and use at least WPA2-PSK protection on the WiFi. Use a generic name on the WiFi that doesn't reveal who's WiFi it is or what device is being used. I like to use names of stores that aren't in my neighbourhood (e.g. Starbucks).
Once you've done all of this, you'll have to reconnect your devices. If they're connected by cable, nothing has to be done. If they're connected via WiFi, you'll have to connect them to the new network with the new password.

Whether you trust the person is up to you. But it never hurts to learn something new by installing the router yourself and it'll give you more safe feeling knowing that nobody is able to do you harm this way.

Big hint: although practicly does it, reading a manual every now and then might help you.
Next time you need a router, ask a store clerk for one that fits your needs (number of wired connections, does it need WiFi, do you want a wired connection that is guaranteed to get better speed than the others [e.g. for gaming, streaming], ...). When you get home, spend some time reading through the whole manual and then start installing the device with the manual closeby.

Is it possible to remotely connect to the Linksys device?

Yes.
Most routers have an online administrator page that can be accessed externally by visiting your public IP address (go check WhatIsMyIP.com too see yours).
It's even easier internally by accessing the internal IP address of your router, which is in most cases either 192.168.0.1 or 192.168.1.1. If these don't work, you can find this IP address by opening a command prompt by opening start and typing cmd (Windows Vista, Windows 7 and Windows 8) or by pressing Windows + R and entering cmd. 

Once the command prompt is open, enter ipconfig and look for the default gateway address under Wireless LAN if you're connected via the Wi-Fi or under Ethernet adapter for LAN connection if you're connected via wire.

If you browse to the address of your router, you shall be greeted by the Lynksys administrative page and you'll have to login. Read the manual or Google lynksys [your model, see box] default login to find out the password.

And see my online activity

This depends. Some routers allow to log the activity of the connected devices. Check the manual or the administrative page of your router for settings like this.
If I were to stalk you, however, I'd connect to your network and run a sniffer. A sniffer snifs all network traffic on the network you're connected on and most of them allow to filter on interesting protocols like HTTP (normal browsing network).

If he knows the password of your Wi-Fi, he can connect to it and sniff your network. Sniffers are tools that listen to all network passing by, even if it's not being sent to you. Wi-Fi is easy to sniff, as wireless data is sent in all directions.

and should I be concerned?

Do you trust this person?

You allowed him in your house and you allowed him to install the device. This means that there is some basic trust between the two of you. Is there any reason not to trust this person?

What can be gained out of doing something like this?
The neighbour might be connected to your Wi-Fi to steal your bandwith, i.e. he downloads movies via your network, so he doesn't have to pay for it. He might also use your network for stuff he wouldn't like to see linked to his own network (for example, criminal activities or surfing to adult websites).

If he's able to sniff your network, he could capture passwords on weakly protected websites (they use normal HTTP instead of HTTPS, which encrypts network traffic). If you suddenly were to see notification of HTTPS-sites being non-trustworthy anymore, although they have always been (e.g. Facebook), then he's trying to sniff the encrypted connection between those websites. If you use other services, like FTP to connect to a fileserver or RDP to connect to a virtual work environment, he could steal those passwords.

By listening on someone's network traffic, it's also possible to collect information about a person's behaviour. Do you check a lot of travel websites, adult websites, cat videos, ... This information could be used to make a profile of you. The advantage of this depends on what it would be used for.

If this issue is possible, how can I defend myself?
If you decided that you don't trust this person, there's a lot you can do yourself.

First of all, undo all possible settings your neighbour made. This means resetting the device.

Unplug all network connections to the device and reset it by sticking a pencil in the hole at the back where it says 'reset'. Hold this for 5-10 seconds. Most devices will notify you of the reset, e.g., with some blinking lights.

Follow the instruction manual of your device to reinstall it. In most cases you'll have to connect a device via cable or via a default Wi-Fi and browse to the IP address I mentioned before.

You'll be able to change the administrative password, name of the Wi-Fi and password of the Wi-Fi. Do all of this. Be sure to use a strong password (8+ characters, capital and small letters and a few numbers, maybe a - or _). And use at least WPA2-PSK protection on the Wi-Fi. Use a generic name on the Wi-Fi that doesn't reveal whose Wi-Fi it is or what device is being used. I like to use names of stores that aren't in my neighbourhood (for example, Starbucks).

Once you've done all of this, you'll have to reconnect your devices. If they're connected by cable, nothing has to be done. If they're connected via Wi-Fi, you'll have to connect them to the new network with the new password.

Whether you trust the person is up to you. But it never hurts to learn something new by installing the router yourself, and it'll give you a more safe feeling knowing that nobody is able to do you harm this way.

Big hint: although this practically does it, reading a manual every now and then might help you.

Next time you need a router, ask a store clerk for one that fits your needs (number of wired connections, does it need Wi-Fi, do you want a wired connection that is guaranteed to get better speed than the others [e.g. for gaming, streaming], ...). When you get home, spend some time reading through the whole manual and then start installing the device with the manual close by.

Is it possible to remotely connect to the linksys device
Yes.
Most routers have an online administrator page that can be accessed externally by visiting your public IP (go check WhatIsMyIP.com too see yours).
It's even easier internally by accessing the internal IP of your router, which is in most cases either 192.168.0.1 or 192.168.1.1. If these don't work, you can find this IP address by opening a Command Prompt by opening start and typing cmd (Win Vista/7/8) or by pressing Windows + R and entering cmd. Once the command prompt is open, enter ipconfig and look for the Default Gateway address near Wireless LAN if you're connected via the WiFi or near Ethernet connected if you're connected via wire.
If you browse to the address of your router, you shall be greeted by the administrative page and you'll have to login. Read the manual or Google lynksis [your model, see box] default login to find out the password.

and see my online activity
This depends. Some routers allow to log the activity of the connected devices. Check the manual or the administrative page of your router for settings like this.
If I were to stalk you, however, I'd connect to your network and run a sniffer. A sniffer snifs all network traffic on the network you're connected on and most of them allow to filter on interesting protocols like HTTP (normal browsing network).

Whether you trust the person is up to you. But it never hurts to learn something new by installing the router yourself and it'll give you more safe feeling knowing that nobody is able to do you harm this way.

Is it possible to remotely connect to the linksys device
Yes.
Most routers have an online administrator page that can be accessed externally by visiting your public IP (go check WhatIsMyIP.com too see yours).
It's even easier internally by accessing the internal IP of your router, which is in most cases either 192.168.0.1 or 192.168.1.1. If these don't work, you can find this IP address by opening a Command Prompt by opening start and typing cmd (Win Vista/7/8) or by pressing Windows + R and entering cmd. Once the command prompt is open, enter ipconfig and look for the Default Gateway address under Wireless LAN if you're connected via the WiFi or under Ethernet adapter for LAN connection if you're connected via wire.
If you browse to the address of your router, you shall be greeted by the Lynksis administrative page and you'll have to login. Read the manual or Google lynksis [your model, see box] default login to find out the password.

and see my online activity
This depends. Some routers allow to log the activity of the connected devices. Check the manual or the administrative page of your router for settings like this.
If I were to stalk you, however, I'd connect to your network and run a sniffer. A sniffer snifs all network traffic on the network you're connected on and most of them allow to filter on interesting protocols like HTTP (normal browsing network).
If he knows the password of your WiFi, he can connect to it and sniff your network. Sniffers are tools that listen to all network passing by, even if it's not being sent to you. WiFi is easy to sniff, as wireless data is sent in all directions.

Whether you trust the person is up to you. But it never hurts to learn something new by installing the router yourself and it'll give you more safe feeling knowing that nobody is able to do you harm this way.

Big hint: although practicly does it, reading a manual every now and then might help you.
Next time you need a router, ask a store clerk for one that fits your needs (number of wired connections, does it need WiFi, do you want a wired connection that is guaranteed to get better speed than the others [e.g. for gaming, streaming], ...). When you get home, spend some time reading through the whole manual and then start installing the device with the manual closeby.