FINAL (hopefully) UPDATE: Well after all the very interesting and valuable discussion, it seems to me as though initial thoughts were correct. From the updated question, I would say that the restrictions are pretty standard for Germany.
My recommendation is that you ignore the noise and the concerns and simply make use of the service. Unless you have some very specific security needs that you haven't shared, using HTTPS wherever possible (which is best practice anyway) is sufficient.
In any case, the other options discussed would all add overheads to your traffic which would use up your 30GB even sooner and slow things down.
There are several things you can do. Provided that the terms and conditions of use are OK with them.
Here are four of the main ways you can protect your Internet traffic from the prying intermediate.
Make sure you always only connect to HTTPS sites
When you use HTTPS sites, the traffic is encrypted between you and the endpoint. Your landlords infrastructure will not be able to do more than examine the destination IP address, port and DNS. In particular, things like banking and health sites will remain secure.
Use a VPN
A VPN in this case is a 3rd party service that encrypts ALL of the traffic (not just web traffic as in 1) between your machine and the VPN host. This prevents any inspection of the traffic at all and it will appear as though you only talk to the VPN destination.
TOR is a way to obfuscate connections across the Internet and is often associated with "the dark web". However, it has legitimate uses as well. Unfortunately, it can add quite an overhead to traffic and may be unacceptably slow. Typically TOR will be used for web browsing, other network traffic would not be affected.
Use the Mobile network
If you are fortunate enough to live in an area with a) good (4G/LTE) mobile coverage and b) an affordable data tariff. Then using a 4G/LTE mobile router may be an option. You can get some staggeringly good data rates.
Don't expect to be free of restrictions though. Many tariffs don't allow device sharing, you'll need a special tariff for mobile data. You might not be allowed to use all services (like VPN's) and you are more likely to have national-level restrictions applied such as the UK's national "firewall".
UPDATE: Without question, the most security would be provided by a VPN.
However, that will only be useful if the landlords network allows VPN traffic. In addition, VPN's also carry an overhead so things like real-time traffic (Skype voice/video for example) and online gaming would be impacted quite significantly.
In addition, VPN's will normally come at a cost though there are some discount codes around that might help.
It is possible to set up your own VPN if you have a server on the Internet to run it on. Most VPS hosts wont allow it but some will as long as you keep it private.
The real question is - do you really need to be bothered? That's why I mentioned HTTPS first. Since this protects your information to sites and since all decent online services already use HTTPS, you might find that this is a storm in a teacup.
UPDATE 2: As some others have pointed out. There are many flavours of VPN. A commercial service will be the easiest to consume but you need to do your homework to find the best for your region. Commercial VPN's can also be relatively easily blocked both by end point and by traffic inspection. Some VPN's require specific ports to be open on the network and these might not be available. Test before you buy. In general, those offering SSL-based or OpenVPN-based are likely to offer more options and be easier to get through any blocks.
Another form of VPN is to use an SSH client such as PUTTY (for Windows) connected to an SSH server (perhaps on your own or a friends VPS). You can throw in a local SOCKS proxy client and then you will have a very configurable private VPN service. Not especially easy to set up though if you don't understand the terminology. Note that many VPS services ban their use for even private VPNs.
Another thing to note is that there are several ways for security infrastructure to spot VPN traffic and therefore block it. Known end points for commercial services and known ports for VPN types are the easiest but it is possible to examine traffic patterns and work out that even apparent SSL traffic (e.g. if using port 443 for VPN) isn't actually.