I downloaded the updater and ran it.
It says "you already have the current version" or something like that.

I start "don't starve" but I am not running the caves version. The updater isn't updated.

I'd like to update the game, please.. GOG?

I've seen, that you already found the solution, but for everybody else:
As of now, GOG doesn't provide the Underground update, yet.

But you can use the manual updater that comes with the game.
Look at that tread for details.