Technically, NYC hacks can go to New Jersey - they just can't bring passengers back to NYC from Jersey (they can't pick up passengers in Jersey).
So - getting to the port is easy. Getting back to Manhattan is harder.
If you want to call a car service "Tel Aviv" is one of the biggest. The average cost is $80 with tips and tolls.
The last time I arrived in Newark and needed to get to Manhattan I just walked out on the sidewalk, saw one of the many car services already there, dialed the number on their sign and said "I'm at terminal 1 door number 4" - I had a car in under 5 minutes, it was $80 total to NYC.
I am the editor, but I also speculate, ask questions and play devil's advocate. I reserve the right to change my mind.
Have not used them but I see these three companies recommended a lot on another Cruise Enthusiast site, in this order or popularity.
Dial 7 cars.
Do Google searches and you will come up with lots of reviews. I think Carmel would be the closest to you..