I still don't understand why you think this fellow might have a tombstone on his grave. He died hundreds of miles from his home and family, and was the prisoner of war at the time. The Yankees certainly did not mark his grave with anything more substantial than a wooden headboard (if they did that), and unless he was very wealthy it is highly unlikely that his family would have come to New Orleans after the war and marked it with any kind of stone (assuming they could have found the grave after the war). Was he rich? Are there any family records indicating that they went to Louisiana and marked the grave? Did he have kinfolk in New Orleans? Unless you can answer "yes" to one of the above questions, the chance of a tombstone ever having been erected for this man is miniscule.
And just for the record, you did specify which cemetery - in your original post, you specified St. Louis Cemetery No. 1.