It’s actually really easy. Mine was totally tarnished. Many metal polishes will work, but I’ve always liked Mother’s Mag & Aluminum polish, and you can get it from any auto parts store. It’s a very fine polish that allows you to be really gentle on coatings like this, but also gives you a mirror shine on bare metal should you ever want to try. Handy to have around.
Anyway, all you need are two microfiber cloths (or one that you cut in two). Just really lightly dab the cloth into the polish tub and polish the yoyo half in circular motions with light pressure. Once you’ve worked the cloth a bit and it’s starting to leave black residue, then switch to a clean part of the cloth and dab a bit more polish. Every 30 seconds of this I use the other clean cloth to wipe away the excess and see how the part looks. This particular job doesn’t take hardly any time at all, you could have both halves back to brand new in less than 5 minutes.
When I’m done I like to run the part under water to get any excess polish off. I found that this may cause the flatter finish inside the cup to want to tarnish a bit, so then I hit that part one more time with a tiny bit of polish and then wipe it away (no water this time).