Look, you absolutely should not touch that original power supply without testing it first. You really need to grab a multimeter and check those voltages—you’re looking for roughly 5.0 to 5.2V DC and about 9V AC. Just a heads-up, don't freak out if the AC side reads a little higher than 9V; that's totally normal when you're testing it without any load attached.
The pinout for the power supply is here:
http://www.vcfed.org/forum/showthrea...-with-no-VoltsOnce you've confirmed the output voltages are sitting where they should be, then you can go ahead and try hooking it up to the Apple II.
Or, if you're feeling a bit more adventurous, just build your own custom power supply, something similar to this setup. I'd highly recommend checking out some videos by John; his stuff is top-tier.
As far as the restoration goes, the absolute baseline is swapping out the capacitors. You can either hunt down individual caps yourself or just order one of those specific recap kits made for the Apple II.
It would also be a smart move to swap the capacitors in the RF modulator while you're at it.
https://www.retroleum.co.uk/c64-capacitorsI've grabbed kits from these guys before for several different Apple II machines, and everything has always worked out fine.
The next thing on your checklist should be the voltage regulators; there are two of them tucked away inside.
The final step is to glue some heatsinks onto all the critical chips.
If you find a piece of cardboard inside the computer case, my advice is to just toss it. It doesn't actually do anything; it was probably just slapped in there years ago as some makeshift EMI shield.
If you run into anything else, just hit me up. There's a massive community on Facebook and a few dedicated forums specifically for Apple II enthusiasts.