The City of New York has its own Census FactFinder: http://gis.nyc.gov/dcp/pa/address.jsp. You can search by "Community District" (these are fairly large) or you can type in a specific address. It will display a map of the census tract, with data on race, income, family configuration, age, and a bunch of other characteristics.
Another tool is http://www.socialexplorer.com. It has interactive maps that will let you search by ancestry. (If you're looking for Irish people, and not just white, you need to search for ethnicity or ancestry.) Click on "Maps" then choose the 2000 Census interactive map. You can select ancestry, then select the specific ancestry you want to look at. Unfortunately, you need to keep zooming in on the map to get to NYC, it won't let you just type in the city.