If the original wiring had only those 5 wires, and the single white wire was going into "-ve H sync and -ve V-sync" as a single terminal, not split, then that's the terminal for Composite sync. If the chassis has no other spots for sync, no separate H- and V-sync terminals, whether positive or negative, and no switches to adjust this, then you have no option but to use Composite sync into that one terminal.
Assuming you're using a Windows PC source here, you can get composite sync from an ArcadeVGA or better yet an AMD Radeon card of the right sort using crt_emudriver. It will be output over pin 13 of the DB15/VGA plug, the same one as would carry H-sync under normal circumstances.
You can also look at combining sync for yourself.
I'm not sure if it's teaching your grandmother to suck eggs, but be aware that most PC sources need a video amplifier when connecting to an arcade monitor or the picture won't be bright enough, though IDK about Wells Gardener monitors in particular. A J-Pac is a good choice for this. It's not the cheapest option but it also lets you hook into the existing jamma setup so you don't have to rewire the cab's existing controls.
Happy to clarify anything, if you need.