you are not doomed...
+1
G07's are fairly common, and there's a lot of places that carry parts. Grab yourself the G07 manual and look up what that Cap is, then get a replacement. Check
bob roberts' site, or
Mike's Arcade, or even
Arcade Shop.
While you're at it, get a cap-kit for it. I know Bob has a full Deluxe Kit, (which may be a bit over board -- but couldn't hurt) that has a cap-kit, a width-coil, flyback, regulator, fuses...you could probably find what you need for C904 -- just be sure to check the manual for the specs.
Since you're handy with soldering, you should be fine with getting this game running in tip-top shape.. I probably have waaaay less soldering life-time experience than you do, (I took half a semester of Electronics class in High School, but switched to Home-Economics 'cause there were way more chicks in it)

but I'm pretty confident with an iron now that I've got a few repairs under my belt. You'll be fine.
That switch toggles the machine On/Off after plugin.
Looks like they rigged it up so that you don't have to bend down to the bottom right corner to flip it on. There's supposed to be a hole at the base of the cabinet in the back there, which the switch sticks out for you to access.
As long as the push button back door switch is taped closed.
From the looks of your picture, the holes it was mounted into looks to be stripped out. Might have to repair that area and re-mount it properly.
You're probably also going to have to get some new cam-locks for the back panel and coin-door. I can't remember the size you'll need, but it should be in the manual. Those can be found at the above links I posted.
As far as the coin door goes..... mixed emotions....
Naw, that looks pretty good. Those cracked Coin inserts can be replaced:
Pinball Resource has NOS inserts you can get (under
Williams Coin Door Parts).
What's that marred-goopy stuff on it? Looks like adhesive from tape or stickers or something. You can get that off with Goo-gone or any glue-remover, found at most hardware stores across this continent. Or, if they're scuffs in the metal, after a good cleaning, take a wire-wheel to it -- put it in your drill chuck, scrape it all away. This is the tried-and-true method for restoring most metal components. Cleaning up bolts is done in a similar manner. Put the bolts in your drill chuck, and run the drill while pressing the bolt down on sandpaper. Doesn't matter the sandpaper grit count, what I do is use some WD-40 for lube, and dip the bolt in that before sanding it down. They come out nice and shiny. A good idea is to clear-coat it after to protect it from rust...
Oh, and for $100, 2 years ago...this is still a freakin' steal! I'm jealous.
Here, check out this site:
http://www.robotron-2084.co.uk/ There's some information on Williams hardware upgrades that you might want to consider later. Things like Ram upgrades (runs cooler , lasts longer) and Battery Mods (uses Lithium batteries rather than those AA ones).
I can already tell, this restore is going to get me fixin' to get on my Robotron Cabaret.