I'm not sure about hunting on Staten Island but you are within an hours drive to some really good public land hunting in New Jersey especially duck and goose hunting along the coast. Check out the wildlife refuge system and also NJ publishes a map tittled open spaces and wold places that shows most of the public hunting land in NJ (It was printed 7 or 8 years ago so the newest lands are not on it).
Heres the link;
http://www.state.nj.us/dep/fgw/

no matter if you plan to trap or not I'd recommend contacting Newt Sterling, in NJ, he owns Snare One trapping supply and he has lived his whole life, so far!, in NJ, better than 70 years of hunting trapping and fishing there, if he can't help you get going in the city then no one can.

Also, I'd look into joining some bowhunting clubs and duck clubs of Long Island ASAP, with literally millions of people and little land available I think it won't be that hard to adjust to Staten Island /LI leases from texas leases.

good luck and welcome!

( formerly posted as slickrock...cant get into my email)

Simon is right, Newt's a good man and a good friend of mine. I just need to warn you....things are a little different in Newts world!

First, Welcome to the site and to NY. Just a couple places I could mention that has good population for after you get set up and ready to go. Weschester County is archery only and has a lot of deer and some real good ones also. I've hunted in White Plains and knocked on some doors for permission to hunt there as it's alot of residential area but most of the land owners there are complaining about the over population because they're sleeping in thier garages, eating thier shrubs etc. I know it sounds funny but believe me, you can get permission there. The next is to scout out the public area on Long Island and call the local DEC office for info on it. It's close to you and has good deer there, just watch the tick's and dress for them. Best of luck !!

Hi and welcome to NY. I was wondering how you liked your governor, Rick Perry? My thoughts is that he would make a great President. ??

I moved to NYC 

 almost 7 years ago and found myself asking similar questions. From SI there are pheasant hunting, Duck, and Goose hunting options on LI but I prefer to head upstate. I ended up buying a house in Putnam Country just so I could hunt and fish.