The Connors Genealogy website has been online since 2001 and contains a large number of transcribed records from New York State and Ireland. Run by Pat Connors, it was one of the early websites to show the potential of volunteer transcription efforts when it came to records concerning Irish immigrants in the United States and record sets from Ireland.
One section of the site that is truly excellent is the list of transcribed Albany County, New York naturalization records from 1836 to 1864. For those that had an Irish county of birth recorded on their petition for naturalization, Connors provides this information, along with their name, age, residence (usually the city of Albany) and date of petition. There are hundreds of such examples. John Hartigan, age 66, petitioned to become a citizen of the United States on 3 November 1846. He was born in County Limerick, resided in Albany, and had left Ireland from Limerick. James Lally was a resident of Watervliet, Albany County when he filed his petition, coincidentally also on 3 November, in 1838. He was born in County Offaly and had emigrated from Dublin.
Helpfully, the Family History Library (FHL) film numbers are provided for each year. John Lally's petition is on film number 1301571. A quick search of the FHL Catalog directed me to the relevant film.
 Albany County NY, Naturalization Records, John Hartigan petition for naturalization 3 November 1846, Connors Genealogy
(http://www.connorsgenealogy.com/Albany/AlbanyIrishBorn1846.htm), accessed 20 November 2016.
 Albany County NY, Naturalization Records, John Lally petition for naturalization 3 November 1838, Connors Genealogy
(http://www.connorsgenealogy.com/Albany/AlbanyIrishBorn1838.htm), accessed 20 November 2016.