A car service will run you as much as $80 each way! The best thing to do is yo find the closest PATH train (Grove Street) which is a subway that will take you right into Manhattan. There are other PATH trainsm but some only take you to the Wall Street Area while others will take you more uptown.
The official line from Royal/Celebrity for getting to the port via public transit is:
Take the PATH train from Manhattan to Jersey City or Hoboken. Outside the PATH Station, take the NJ Hudson-Bergen Light Rail train to Bayonne. Get off at the 34th Street Station in Bayonne. You are now directly across Route 440 from the Peninsula at Bayonne Harbor, which includes the Cape Liberty Cruise Port.
Call one of the three Bayonne cab companies to drive you to the cruise port, which is about two miles east of the Light Rail line.
Two trains and a cab on top of whatever it takes to get to Penn Station seems like a lot of work to me (lazy). Is there a better option? I was thinking maybe NJT to EWR and then take a cab from there. How workable is this?
A Bad Day At Sea [with power] Always Beats A Good Day At Work
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014
Star Clippers: Royal Clipper 2015