You might want to consider looking into the Woodlawn section of the Bronx. This is a very well established neighborhood with a large number of Irish restaurants and pubs. There are also many Irish immigrants living here, both legal and illegal. Well, it ain't illegal unless you get caught I guess ;-). (
http://en.wikipedia.org/wiki/Woodlawn%2C_Bronx)
There is also a neighborhood in Queens called Woodside, that is popular with Irish immigrants, but I am not too familiar with it.