How do I go about fixing an apartment buzzer?
I would report it to maintenance/landlord, it isn't something you installed to the unit so it isn't your responsibility to repair it.

Plus, the issue isn't even necessarily at the end in your unit, there could be a problem with the button at the front door, wiring issue between, etc.

Typically to test wiring issues you need access to both ends as you will need to loop one end and test conductivity from the other.

Aside from if the buzzer device itself went out, I'm not sure what you could fix on just your end, and that would require a part replacement that you don't need to pay for.