my airport's ability to connect with internet is broken.

Not a lot of details there as to your current network configuration with your AirPort base station, including which exact model your base station is.

I'm assuming that it is a Time Capsule.

The basic steps to continue to use it for Time Machine backups would be to connect it directly to your other router by Ethernet. Reconfigure the TC in Bridge Mode, and then, configure its wireless radios to use the same network name & security password as your other wireless router. You can then use the TC for TM backups as before.

just be a backup disk wirelessly?

BTW you do not need to use the TC wireless if your main router is working better .. instead of setting up the wireless with same SSID (or different if you want to keep them separate) you can just turn wireless off on the TC and use whatever wireless your main router is using.. I would suggest this SHOULD be done if you have main router and TC parked close to each other. Using two routers with wifi even in the same room is not useful. Pick the better of the two and use that one.