Since that engine has a cast iron intake manifold, it will be too old to have a tag riveted to the top of the cooling shroud. The number would have been on the cooling shroud covering the side of the cylinder to the left of the picture when facing the intake manifold, #1 cylinder. Well, it appears to be a hybrid since it has plastic cooling fan, so maybe had a tag. With a cast iron intake manifold, it should not have plastic cooling fan. Of course it is an "L" head so will not have numbers on a valve cover. Can't tell if it is a 40 series or a 42 series.
Vertually any B&S starter will work on that by changing the end mounting plate. The mounting plate for these often wore the top bushing oblong shaped letting the armature rub on the windings so check that on any replacement. While a replacement bushing is not offered by B&S, you can get one close at farm supply or hardware and easily adapt it for a replacement.
I can send you a Service Manual covering the engine IF you like, address below put in proper format and remind me what you want.
Walt Conner
wconner5 at frontier dot com