There's the Charles B Wang Center summer internship. I believe it is called Project Ahead? I can't remember and it's been 6 years since I interviewed.
NY Downtown hospital has a pretty decent volunteer program but again, I haven't been in 7 years.
Community programs in NYC tend to offer better volunteer opportunities than private hospitals like NY-Presb, Tisch, Sinai etc. Those programs will probably make you a hospital guide or some sort of file ***** (read: very little patient interaction).
PHC is one of the BEST programs in NYC for patient interaction and learning about medicine. By the way, age will NOT hurt you. They choose candidates of ALL ages. I participated back in 2007. I'd say a good number of PHCers that summer were recent freshman/sophmores. There were a number of college graduates, the oldest being 26 at the time (all of whom eventually went on to do a post-bac and are now in medical school). Heck, they even took me and they KNEW I was entering medical school immediately after the program ended. You just need a solid reason for being in the program. Good luck!