I can speak to my experience here as an undergraduate. There are various bus routes in Suffolk County which will take you to shopping centers (when I left there was a student rate during the week and it was free for students on the weekends). There are also a couple school-sponsored buses several days a week which will take students to local shopping plazas, running every 30 minutes I believe. I can't speak to the necessity of having a car during clinical rotations at the affiliate hospitals, but I can say a car will at least be very helpful. You may be able to manage, but managing your life around bus schedules is not always easy.
As for apartments, $1500 for a single-room, no-utilities apartment is definitely on the higher end and only that much because Port Jeff is a nice area. I was living in a nice two-bedroom apartment with all utilities beside AC for $1500, and that was considered expensive.