A soft toothbrush under running water is what I plan to use on SS, as well as on the link bracelet. I'm imagining that this summer there will be sunscreen, dead skin cells, dust, and sweat to be cleaned out of little places.
Did I see somewhere that you can clean the exterior with some jewelry-cleaning solution? You might look that up and then hit a jewelry store. My mom used/uses that stuff for I think rings and necklaces made out of stainless steel, maybe silver and gold. I would imagine you can find a brush made for something that small and maybe that solution will help remove it. But I certainly wouldn't go submerging it in anything. I'm guessing you knew that, but I just wanted to write that so you wouldn't lose a good watch.