You wouldn't use a router to connect to the internet.
If you have free wi-fi, then you would just add that to whatever computer has access. You would need to know the NAME which should appear as one of the local wi-fi connections and if it's secure (which it should be) you'll need the password as well.
Example:
1. Start computer
2. Click on network connection and "add wireless network"
3. Choose the proper name "Apartment network name" for the shared wi-fi network, and
4. Use password if needed
You'd then be connected to the INTERNET through the apartment's modem. (It's like accessing a Starbuck's network)
A router can connect different devices on a local network if you need to transfer files between them but otherwise it serves no purpose in your case.