Find a Grave has him buried here:

Bethlehem Cemetery, Spotville, Columbia County, Arkansas, USA

https://www.findagrave.com/memorial/search?firstname=james&middlename=taylor&lastname=ainsworth&birthyear=&birthyearfilter=&deathyear=1864&deathyearfilter=&location=&locationId=&memorialid=&datefilter=&orderby=&page=1#sr-87035643