None of the decent apartment complexes i know are close to FPR. Give Sabal Chase a try. They have contacts to a lot of flight students and do not require the usual social security stuff etc.
You can have also a look at Vero Beach, but i am not that familiar with their apartment situation.
You definitively need a car in FPR. The airport is quite off the hook.
Tip, stay away from all these "Avenue A,B,C etc." streets when you cross US1. They have a quite aggressive neighbourhood.
Sabal Chase is right at the H95 exit (together with the turnpike). from there you drive approx. 1km east and as soon you see walmart to your left take the turn and follow the road.
hope it helps a bit
cheers, masit
(was there 1 year as cfi, if you have questions regarding FPR, drop me a mail)