I searched on Arlington Memorial Cemetery (Google search), which is a location of Civil War Veterans gravesites in Kearny, Hudson, New Jersey. I then found a map of Arlington Memorial Cemetery with nearby cemeteries noted. On the list of nearby cemeteries is Mount Olivet in Bloomfield Township, Essex County, New Jersey. LINK: http://www.histopolis.com/Place/US/NJ/Essex_County/Mount_Olivet_Cemetery/313575204http://www.histopolis.com/Place/US/NJ/Essex_County/Mount_Olivet_Cemetery/313575204
FOUND on Ancestry.com, New Jersey Deaths and Burials Index, 1798-1971 Name:Patrick Ford Or Fay Birth Date:abt 1842 Birth Place:Ireland Death Date:26 Jul 1896 Death Place:Kearny, Hudson, New Jersey Death Age:54 years Occupation:Laborer Race:White Marital Status:Married Gender:Male FHL Film Number:589805
So "your" Patrick was born about 1842 in Ireland. I did not find him in a US census in Kearny, Hudson, New Jersey, or in any New Jersey county. But he was staying in the "old soldiers home" in Kearny when he died, is that right? So Kearny would not necessarily be where he had previously lived.
Also he could be in the census with a spelling of his name.
There are so many Patrick Fays in the census and in Civil War records that at the moment I haven't located which one is "yours."
I hope this helps. Thank you again for all you are doing for this worthwhile project!