Here is the Mega bus website.
It will take you from New Brunswick to NYC Bus Terminal (about 30 min. trip) for $8-10 one way. There are small van-buses that'll take you from Union City (directly across the Hudson River) to the bus terminal for $6 one way. The Union City bus stop that we use is in front of the Post Office on Kennedy Blvd. On the return trip you catch the bus again from the curb outside of the bus terminal (ask the driver where when you get off at the terminal).
New Jersey transit is the most common public transport from NJ to NYC:
All that being said, the Staten Island ferry is a fun way to get to Manhattan and back--and it is free!