Second Street doesn’t really have many video games from what I’ve seen (granted I live in the country, so it could be different in Tokyo). Small hole in the wall shops are always the place to go.

I used to sell video games professionally, and I used distilled white vinegar and a q-tip. Just dip the q-tip in vinegar and wipe the pins, then dry it with the other end. A bunch of black crap will come off - keep doing this until it’s clean! To open it, you need I think a 3.5 mm game bit - you can buy them on ebay for a few bucks. You can also buy the batteries with the metal bits already on to solder it back onto the board. Make sure the polarity is correct!! I believe most gameboy games take 2025 batteries but you’ll wanna double check, I think a couple take 1212s or 1216s. Hope that helps :)