Hi Bob, what year was Patrick born? What places did he live? What was his wife's name? There are quite a few Patrick Fay's of the right age to fight in the Civil War, and some are in New Jersey in some census years.
Most people are buried: 1. in/near the town/place where they died 2. in a place where they lived previously 3. in a place where a wife, child, parents, siblings, etc are buried
You hold these clues, so maybe you'll have better luck with your post if you share them.
Here'sFind a Grave memorial to an Unknown Fay buried at a Mt. Olivet Cemetery in Bloomfield, Essex, New Jersey: