Eureka!!
John W. Shoptow [sic], Company F, 1st Arkansas Cavalry Battalion, born in Kentucky, died October 15, 1863 in hospital at Columbus, Miss., left no effects, no other records
M347: Unfiled Papers and Slips Belonging in Confederate Compiled Service Records
Note, this is the one and only card found in Miscellaneous papers for him
There is only one muster roll for Company F found in the records, this being for the bi-montly period Jany/Feby, 1863 when they were stationed at the end of the period at Camp Donelson, Miss. This would explain why his records have not been found and would suggest he enlisted sometime after February, 1863. The only muster roll surviving appears to be that dated February 15, 1862 at Russellville, Ark.
There do not seem to be any records extant for the Columbus, Miss. hospitals
See also:
http://www.couchgenweb.com/civilwar/stirmdx.htm
http://civilwarroster.com/cw/rosters/stirmans.html
http://www.nps.gov/civilwar/search-regiments-detail.htm?regiment_id=CAR0001BC
https://www.familysearch.org/learn/wiki/en/1st_Battalion,_Arkansas_Cavalry_(Stirman's)_(Confederate)