I searched the Irish Family History Foundation's database for Co. Armagh. Rcds B4 1840 are sparse.
For William Smith B 1837, I found 0 rcds. The closest were in 1835 and 1840.
For James Smith B 1815, the closest I found was in 1814.
For a James Smith marriage abt 1835, I found 1 in 1833. A view of the M rcd SHOULD show the wife's name. This will validate/invalidate the rcd. If validated, it might give his exact age - which can be used to validate/invalidate the B rcd.