There are a number of approaches you could consider. None of them involve using the browsers password store.
Firstly, you should consider using a dedicated web-based password management tool such as LastPass. Most importantly in this case, this has a one-time password feature that allows use of LastPass on untrusted devices.
You can also add multi-factor authentication options to this for added security.
This doesn't make you completely secure though and you should always assume that an unknown computer is compromised so restrict what you do carefully.
In general using a mobile phone will be more secure than an unknown computer.
UPDATE: LastPass protects you on an untrusted computer because you use a single-use passcode (that you have to have generated in advance). That gets you access to a web page, your "vault". On that page, you can directly launch a password protected site without having to manually type your id and password. For most sites, this will prevent any key-logging or traffic interception of your password.
It goes without saying that you get no protection from this on any site not using HTTPS.
It is also important to note that this only protects against capturing of your passcode. It does not prevent capturing of other information, nor would it help protect against certain malware designed specifically to attack financial sites such as personal banking. An untrusted computer is always untrusted. However, it may be enough for you to grab an image or low-value document from your iCloud or other cloud service or to log into your webmail to check non-sensitive information.
It might also be helpful for me to note that I don't trust LastPass for everything. I also use a stand-alone system called KeePass for really high-value passcodes. This is still cross-platform (e.g. both desktop and mobile) and with an easily copyable file database but can be kept offline or hidden away somewhere. So you could use that as a "phone vault".
I still wouldn't access finance and other high-value sites from an untrusted computer though.