I do not have personal experience with this issue of living in NJ and getting ticket in NY.
I can give you a bit of insight on the NY State Trooper ticket procedures. In MOST cases you need to show up to get any reduction. They usually plea bargain with the actual ticketing officer and they will reduce you to half speed. So if you got a ticket for 20 over, they make it 10. But you pay heafty fines of $100-$300 (in that range) with the court costs either way. In NY State courts you CANNOT buy points down, at least as of my last ticket a few years back.
I know NJ has some reciprocation with NY. I found this Website, you may want to call and give it a shot. But it gives some info on your situation:
http://www.redlichlaw.com/traffic/ne...g-tickets.html
Good Luck!