So, the Briggs and Stratton engine is 130 cc, an example of their first design generation, with a horizontal crankshaft, a Pulsa-Jet carburetor, a flange-mounting built-in reduction gear, ball main bearings, and a rewind starter. It obviously does not have a reduction gear now, so that has presumably been removed. That engine is almost identical to the correct engine for your mower, which was an 80202: it had plain main bearings and no reduction gear. Both 80232 and 80202 are 3 hp. That was the best original engine ever fitted to the SB45 as far as I am aware. Your engine was made on 22 February, 1979, which means it is probably much the same age as your mower. If you get rid of the Finsbury tag and paint it orange, it will be pretty much indistinguishable from the original engine.
I suggest you undertake a diagnostic program to find out why it doesn't start, use it a bit, then decide whether you like it.
Before you try to start the engine, check the oil level in the sump. It should be full up to the top of the filler plug's thread. The engine has a manual choke, and will not start unless you use it. I hope you haven't been using it up until now, or you have probably flooded it and made it more difficult to start. The standard starting technique is to set the speed control for maximum, close the choke (pull it straight out, as far as possible, which is about 1/4" or so), pull the starter cord about twice, and if nothing is wrong, it will chug a couple of times and stop. You then push the choke right in (which unchokes it completely) and pull the starter again. If nothing is wrong it will then start, almost certainly on that first pull, and run sweetly until it runs out of fuel or you move the speed control to the stop position.
If you do that and it doesn't start, report back on what it actually did, and we can go on from there.
