As long as he bought a digital or analogue watch with solar charging then even that shouldn't be an issue. The intensity of sunlight on the moon's surface on the light side is around 37% stronger than that on earth due to the lack of atmosphere, so it would easily charge a watch's integrated battery. The main issues would most likely be the high temperatures which can reach around +127 Celsius and the strong UV radiation, so as long as he chose a watch that could handle both of those he should be ok.
It would certainly be more an issue though if he wanted to visit the dark side as the temperature can drop to around -173 Celsius, but most importantly there would be no sunlight at all (or perhaps only a very very small amount from starlight received from the observable universe) so the battery would not charge unfortunately.
Other than that, and assuming he could get NASA, ESA or Roscomos (depending who he was travelling with) to flight qualify the watch he should be good to go.
Excellent