Like Wilba says use the PATH train to World Trade Centre or Newark Liberty Airport Express Bus to Grand Central, New York, this service runs from the airport and you can find the times on the net. Lots to see when you are there at that end of Manhattan, again look it up on the net. When you come back do be extremely careful around the Penn Station area of Newark it's very dodgy after dark.
Thank you for that question, my husband and I will be flying from Birmingham a day before the cruise goes our and hoped to spend the afternoon and early evening taking in some sights in Manhattan. We had no idea how to do this, so depending on hotel we are allocated we will look at using the tube.
20 minute cab ride or Newark Penn Subway (tube) goes to World Trade Centre (Ground Zero). Google a map of Manhattan which might help.
Take the hotel shuttle to airport p4 up the escalater to air train every 4mins. take this to train station then train to manhatton costs $15 dollars one way per person it was the last stop when we went on friday 21st nov. very easy well worth it takes about 1hr. from hotel to new york. when you come back take the shuttle back to the hotel they are every 15mins. no charge .