Are you looking for a battery replacement or a repair for an old analog springs and gears watch? There are some stores in malls that make battery changes, even street side watch repair kiosks can do that for cheap change.
I guess it depends on what you need repaired. Many years ago I brought a watch to a watch sales and repair shop in the back of the fish and vegetable market on South Pattaya Road, roughly across from Wat Chai Mongkhon and not far from TutCom.
Malls have upscale stores selling watches that offer repairs. There are many malls in Pattaya. When I did a quick Google search there were several places listed. Depends on what you need done. If you have a very expensive Rolex type watch maybe getting the repairs done back home would be safer.
If you are passing through Bangkok, there is a very good watch repair shop next to the Emporium Mall on Sukhumvit Soi 24. It's approximately 50 meters off of Sukhumvit on the right hand side.
There are plenty in Pattaya. Most of the jeweler shops selling watches in the nice big shopping centers do and are authorized to do repairs by the big watch brands for a fraction of what it would cost back home.
The place I go to is on Soi Boukow (10baht baht buses run both ways). Google for a map. Head toward the T-intersection with Pattaya Klang (where Soi Boukow ends), about 20 metres before on the right hand side is a small repair shop. Only place I go to. I've had 1 watch repaired & cleaned. Others have just been battery replacement.