Check out this website, we have used them a few times when sailing out of Bayonne, years ago.
Check out the hotel below, it wasn't on their lists of hotels when we used them last, but it is something to consider since you are going to need 2 car rentals, and I can't think even for 1 day is going to be cheap in that area.
Your total would be like $365 for hotel( 1 night)23 days of parking, they will shuttle you back and forth from cruiseport also, plus you get to sleep in the hotel the night before, no worries of car trouble on your ride that day of the cruise.
Hotel has amazing reviews:
Clark Holiday Inn - Cape Liberty Park Stay and Cruise for 3 or 4 Includes round trip Cape Liberty Cruise Port SUV Service for 3 or 4 with parking
36 Valley Road
Clark NJ 07066
Distance from Airport:
14 Miles from Newark Airport, 20 Miles from Cape Liberty Cruise Port
Airport Shuttle Hours:
Cruise Port Shuttle:
Includes round trip Cape Liberty Cruise Port SUV Service for 4 at the times of your choice
includes 21 days of free parking
$10 per day after the 21 days included in our package
One room for one night only for up to 4 per room
is not included but available at the hotel.
Friday to Sunday(January)
$ 345.00 inclusive
Good for up to 4 people per room unless
Other wise described above.
(final cost, no added fees)
Room Type & occupancy:
One room with two double beds.