Its possible to buy commercially made watches that do all that the OP needs.
Mainly made for athletes for running or biking.
Garmin is the most popular.
Very hard to make something like this yourself and keep it small and lightweight.
Even with the commercial offerings, the GPS is what uses all the power and battery life is around 10 hours
with the GPS function on , much longer with it off.
The watches use LIPO rechargeable batteries.