How to remove Bathroom Ceiling Fan
Most of those fans have brackets attached to the sides of the housing that extend and are nailed into the studs. Since it's the motor making the noise, try unplugging it from the housing and removing those silver screws. You should be able to remove the motor and fan impeller. Now try to clean it up and dab a few drops of oil on the motor bearings.Re install it and see if it's better.
Once the motor's removed, you'll probably see the mounting bracket sleeves or nails going into to studs that hold the housing in. You could then try to work those lose. Since this is an apartment, check with the landlord before doing any work. He might even have a replacement motor that you can just screw back in.