I find a J. S. Cunningham in Co. D, 1st Tennessee Cavalry but the 1st Tn Cavalry was not in Virgina during the war. Most of the regiment was captured near Shelbyville, Tn. in late June 1863 and 292 of them were shown as POW's at Camp Douglas, Illinois on a report dated December 26th, 1863. Co. D was made up of men from Major H. C. King's 1st Kentucky Battalion.
Where did you get the information you have given here? There seems to be some confusion involved with either the person or the unit he was in or both.