The case number would be on the inside of the case back and is 3 or 4 numbers, some with a single letter in front. I'm not sure when they started using the case numbers, the earliest example in my own collection is 1962 and none of mine from 1958 have it, so I would guess sometime between 1958 and 1962 is when they started using the case numbers.
You can download the ABC Catalog and Supplement here. There is a section at the beginning explaining how to use the case numbers to find the correct Bulova part number. If your watch does have a case number and you look it up, it will have a line showing all of the correct Bulova part numbers, but if it has a split stem it will tell you to go to the "Split Stem" section. For example, case 622, under stem it says "*(Split Stems)", so you go to the Split Stem section (starts on page 110 in my copy) and then look up the case number there. For 622 the correct female split stem is found on page 117 and shows the part as 16C-11AC 11.28 MM. Unfortunately, this section seems to be incomplete and for some reason it doesn't list all of the split stems in this section that it should. For example, case 2981 shows as a split stem and tells you to go the Split Stem section in the catalog, but there is no entry there for that case.