Roster of Confederate Soldiers, 1861-1865 shows:

Brown, James M. MS 36th Inf. Co.E
Brown, James M. MS 44th Inf. Co.F
Brown, James M. MS St.Cav. 2nd Bn. (Harris') Co.A

36th MS Inf Reg Organized early 1862 and was at Vicksburg; surrendered 7/4/1863; Paroled 7/4/1863; exchanged 9/12/1863; surrendered 5/4/1865.

44th MS Inf Reg was NOT at Vicksburg.

MS St Cav 2nd Bn (Harris') Co A was NOT at Vicksburg; spent most of it's time around Atlanta.

It looks like the person you are looking for is:

Brown, James M. MS 36th Inf. Co.E

soldiersearch@ec.rr.com or 910-686-9591 can provide you with service records.

Consider contacting :
Mississippi Department of Archives and History'
100 South State Street
POBox 571
Jackson, MS 39205

601-359-6876