The long path / palisades interstate park trail (goes parallel to river road on top of the cliffs north of GWB) is a hiking only trail. I have ridden it once but it is illegal and they do mean it. It has the most no biking signs per trailhead than anywhere else I've seen, and the roadways get patrolled often by rangers, police and security guards. I think this is because of its proximity to GWB. I have been yelled at, not for riding the trails, but some other stuff.
The closest thing that is legal is in NYC at highbridge - it is not the most awesome, but I guess it is something and impressive that there is actually something inside NYC:
There are also public transportation options to Blue Mountain, Ringwood, Long Island and most other places, provided you are ok with pedaling from train station to trailhead. Let me know if you want more info on that and I'll dig it up.
Depending on how long it takes you to get on the highway, you're really not more than 50 min from Ringwood or Mahlon here. Both huge networks of trails with a shit-ton of MTBNJ riders able to show you the ropes.