Earliest cite OED has for on the nose is this one.
1883 Sporting Life 20 May 1/3 He hit the ball fairly on the nose, sending it clear to the right field fence.
Baseball perhaps? (N. Amer. origin according to OED.) There’s also a separate entry for the horse-racing sense of betting on a nag to win. That has a 1934 cite and seems to be a derivative of by the nose, at least OED lumps them together.
Note that the OED (2003 entry) has a horse winning “by a nose” from 1851. So that form and sense are older and from horse-racing and may have influenced on the nose (it almost certainly had an influence on betting on the nose).
I wouldn’t rule out boxing—antedatings may always be found—but the evidence so far is for the baseball origin.