Here is some info for you: Collecting and Shooting the Swiss Karabiner Model 31 (K31) - HISTORY
The average K-31 has marvelous metal, and a stock that looks like a reservist stood around with the butt of the rifle in snow for 2 weeks a year. Because that is exactly what happened.
Get as much of the Swiss Milsurp GP-11 ammo as you can. It is great stuff, and when gone, will not be seen again.
Rifle will likely shoot better than you. You can add a no gunsmithing scope mount. The unique shape of the bullet makes for a REALLy good long range rifle. For easy windage adjustments, (from sight is in an angled slot, you push it forwards/backwards) visit an industrial supply store, and buy a shaft collar. (Metal ring with setscrew that slips over the end of a rotating shaft to hold it in place) Get one big enuff to fit horizontally over the front sight. Turning set screw will push the front sight blade, allowing fine movement without beating on the sight.
You may find a slip of paper under the buttplate- name and address of Swiss citizen it was issued to. In case you forgot, and left your rifle on the street car, so it could be returned to you. Seriously.