Saturday morning is a good time to drive into NYC. It’s truly easy at that time. Lots of parking in the area as well. I’ve parked on the street at NYU on Saturdays, but you’ll also see signs for parking all over the place. App is great but, on a weekend morning, probably not necessary. All of these are nearby: https://www.nyu.edu/about/visitor-information/parking.html
If you want to do part of the trip the evening before, keep in mind that you will be tired! Maybe just drive the 1.25 hours to Syracuse and stay there, do the rest the next morning. Binghamton is another logical stopping point. I would not plan on driving any more than that after a full day audition.
Last year, my S had back-to-back auditions at Eastman and Syracuse. We drove to Rochester (about 6 hours from New Jersey), stayed over, did the Eastman audition, then drove to Syracuse and stayed right near the 690 to I-81 turnoff (there are a Crowne Plaza and a Marriott there, among others). I was beat! The drive to NY after the Syracuse audition was a little over four hours. If you can make it to Binghamton (tons of hotels there as well), you’ll cut it to three hours.
I hope you are planning to stay at least one night in NYC as a treat after all that driving!