I have recently rented an apartment in Bronx, NY and my office is in New Jersey. Will I be paying NYC income tax ?

Yep. Residents of NYC pay resident income tax regardless of where the income comes from. Of course, you also pay NY state income tax, and NJ non-resident tax.

Does that mean additional taxes meaning paying state taxes two times

Also one more Q.. I can choose to go the office in NYC rather than NJ, while still living in NYC. Which is a better deal from a tax perspective ?

First, regardiing working in one jursidiction and living in another -- you do not get taxed twice on the same income, but you do et taxed at a rate that is determined by your total income. You figure your taxes first on income earned where you work, which you pay in that jurisdiction, then you figure taxes on your total income for your home state, and take a credit for taxes paid elsewhere. In your case this would work out approximately as follows:

1. Pay 5.5% or so to NJ on your NJ wages (this depends on your tax bracket)
2. Pay an additional 1.3% or so to NY State on your wages (because NY State's rates are that much higher than NJ's) plus the full NY State rate of 6.85% on any other income, such as dividend, capital gains, etc.
3. Pay another 3% or so to NY City for resident income tax.

These percentages are approximate of course, but you get the idea. You won't save (except perhaps in commuting costs) if your job was in NYC versus NJ, as long as you live in NYC. You would save substantially on taxes if you were to live and work in NJ.