Quite a lot, but mostly out of fear of being wrong; I believe that identity is such that I will not wake up in the spaceship, and given that I'm correct I would care very little about the fate of that copy of me, but my beliefs about identity are based on strong intuitions that aren't soundly philosophically based and merely seem "obvious". I'd guess I care about 5% as much as I would care if I were to be placed in the spaceship, although "5% as much as if I was to be put in a torture-can and shot into space" is really abstract and I have no idea how to convert that to actual amounts of caring.