I would think $4K - $4.5K is pretty generous budget for a 1br in East Coast area. You can probably get something brand new and well-furnished with that budget, or you can get a 2-3 br apartment (not entirely brand new but reasonably good condition) for that budget.
Maybe you can zoom into your personal preference (since you have a more generous budget) and more recommendations can come in? Things like, do you prefer to be near a train station, near the beach, city area, near to office, etc.
Rent deposit wise, generally, its 1 month deposit for every year you are leasing (so 2 months deposit for 2 years lease).
Setting up an apartment: For me, I used "Starhub" for my mobile, internet, cable, phone lines (which is free actually, when you subscribe to internet). To me, its more convenient dealing with 1 company than tracking down the cheapest options.
For most rental places, you would need to apply for your own utilities (ie. electricity & water). The
property agent can help you deal with that (or mine did!), and you would need to place a $300 deposit (if I remember correctly) for
condominum and private apartments.