In the Nebraska resident military roster of 1891 it lists a Henry Fleming of Atkinson who served in Company B, 43 Ohio. A Henry Fleming who served in that unit received a Civil War pension Appl 61,866 Cert 85016.
You can get a copy of his pension from the National Archives in Washington, DC for $75. While a bit pricey and the genealogical data it contains is uncertain, it may have some of what you are looking for. As I understand all the pensions will eventually be online, but I don't know the timeline. Some are already available online at Footnote.com for a fee.
The connection I found seems plausible, but not certain. His obituary may prove helpful and hopefully he had at least a few surviving siblings. If he outlived all of his siblings their obituaries might list him.