As save-games and configs of my GOG Linux games are normally(!) outside of the installation folders (found them in my home-folder, under .local or .config), I simply install such new version .sh file into the same locations (overwriting the existing installation). But I do not know for sure, if this is the best solution in all cases.
Anyway you can easily try it by yourself after backing up your game-folder (simply zip it). This way you will not lose anything if overwriting game files will mess something up.
EDIT: maybe it is worth finding the save-game/config folder and back it up, too. Just in case the new version messes with config files.