Talk to the cemetery director, if there is one, they normally have the proper forms and will help with the paper work. One of the requirements is that Cemetery management must agree to allowing a marker.
Generally speaking, the rules were that they would not provide one if the veteran already had a marker of any type. However, I have seen this violated many times.
If this is an old cemetery without a staff, you should be able to get the nearest Sons of Union Veterans to assist in placing and performing a ceremony.
As to physical descriptions, I would hazard to guess that men are lousy at describing a persons physical description. Personally, If I were given those two descriptions at different times, I wouldn't recognize it.
Best wishes,
George Martin