[Question] Cleaning/polishing a watch

I have a watch that's fairly dirty just from regular wear and I wanted to clean and polish it up. For cleaning, I'm assuming something like a lens cleaner on a cloth, but I also wanted to polish it a bit to bring back the shine. For that I was thinking a Dremel with a buffing pad and polishing cream. But then I started to think I might mess up the finish in some way (i.e. accidentally hitting the AR coating or the like).

Is this the best way to polish it or is there another way?