Around what area in NJ do you need to get to? That really changes the equation here. I wouldn't hesitate to use NYC and NJ public transportation on a weekend from 10PM - 1AM.

First, Take the AirTrain to the LIRR Jamaica station. Take the LIRR train to New York Penn Station/34th Street in Manhattan. If you are staying in or around Hoboken, Harrison, Newark, or Jersey City, then you can take the PATH train into NJ 24/7 and can Uber from within NJ, as it'll be cheaper.

If you are going somewhere not on the PATH line, from New York Penn Station, catch a NJ Transit commuter rail to the closest station that you need. Then you can Uber from there. Most NJ Transit commuter rails end for the night at 1AM.