No, sorry, I'm not familiar with the area by the airport and Bayonne. To be honest, I only drive that way to go to Newark airport, but if you were looking for something on the nicer side, I might try the Marriott Newark Airport, or for less money the Courtyard Newark Airport. I don't know if they provide a shuttle to the port, but it may be worth looking into, as I know Marriott hotels are pretty nice speaking from experience traveling with family. The Marriott is more expensive, so if you're on a budget, I would go with the Courtyard. Also, Marriott hotels may provide shuttle service; you may want to check on this as I am not sure.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)