Just do a crash pad. Most of them are about $175-225 a month. No point in renting an apartment in Jersey unless you are dying to make it your permanent residence so you can pay a ton in taxes.
If you're going to be on first year pay, you will most likely only be able to afford the questionable areas in Jersey to rent.
Public transportation isn't bad. Trains and buses run a lot. Cars will be broken into if you are gone on a trip if left at your residence if you're in a "shady" area.
Can you commute for now or are you deadset on moving to this base? Also, are you going to be bounced around in bases?