Good work on the C64
Anyway it's these caps that are the likey culprits:
Probably the blue ones too, but I'd guess it's those larger orange caps that are on the +12V line. They're used to filter out noise, which is great until they fail and go short circuit. Being in parallel means you wont be able to find the specific bad cap without removing it from the board first.
What you should
do, is identify all the caps on the 0 ohm rail (+12V), then desolder / test and replace where necessary.
do, is identify them, then bend each one back and forward until it come soff, until the resistance of +12 increases (usually jumps from 0 ohms to something like 1500 when you get the right one).
So far I haven't had a machine suffer from them being partially removed - not many devices use +12 and those that do usually have their own power filtering. In saying that, if you have the time, putting in new ones to keep it up to spec is desirable.
It's a good plan to do this first, just to make sure the +12 or -12 line feeds are not required for the memory controller (or similar) to operate correctly.
It's a standard AT connector, not the usual Wang setup where there is square plugs. So that plug provides extra grounds, +12 (yellow), -12, and Power Good (the last pin, which nothing pays attention to).
RAM chips looks like the usual array of 41256 or 44256 (not 414256) chips (even if the number is different, I'd wager that's what they are).