On weekends, the Harlem Commuter line out of Grand Central Sta. Has a stop right on the AT only a mile or two from the CT border, near Pawling, NY and Wingdale. The Pawling station is 3 miles south of the trail, so you coulkd take a biike on the train and get there. The AT leaves NY/NJ at Delaware Gap. Frequent buses from NYC stop at Delaware Gap. The bus stop is less than a mile from the AT. Public transportation might get you to Bear Mt., too. The AT in NY/NJ is only 30-40 miles outside of NYC, yet in miles of wilderness, small farms, and a wildlife refuge. Beautiful. The trail goes along the northern border of NJ, so you are in and out of NJ and NY.
As an AT section hiker, I have used my bike from New England to the Smokey Mts., and never had a problem. I lie it flat on the ground under a small tarp and chain to a tree. Just be sure you remember where you hid it. In fact I hid it in the bushes right near the AT stop in NY I referred to above. It was there for a week. I also left my bike at the top of Clingman's Dome in the Smokies. It was a great ride back down! That was probably my shortest ride after a hike, the longest was 80 miles in Maine!