go look at nynjtc maps at any outdoor store, i recommend harriman/bear mtn or catskills maps.
there are many short hikes to shelters/leantos. you can sleep in the leanto or set up a tent within 150 feet.
hunter mountain has 3 nice shelters and a loop hike 8 miles.
harriman is 45 minutes from the city and has tons of options. i recommend tom jones shelter, big hill, bald rocks, west mountain, all relativly close to trailhead.
click on viewpoint for discussion forums for other info etc