One possibility is that he still thinks mostly like an ordinary human, but accepts the incredibly bizarre as normal without a second thought. Assuming he's actually enjoying himself somewhat, this is quite plausible.
So, he still has worries, desires, and concerns like most people, but all the weird things he's seen flow naturally into it, as though there wasn't anything strange about them. You know, something like, "Hm, what should I have for lunch today? Guess I'll figure it out after I check if someone's done something about that hole in reality yet."
I've experienced a very mild version of this, myself; after being exposed to so much weird mathematics, I'm not fazed at all by the latest crazy news from physicists. The universe is actually a hologram, or there are hidden dimensions of time? Okay, sounds plausible, could be a lot weirder.
Your handling of this character should also depend on how serious the tone of the story is. If you want to play up the elements of horror, he should be more psychologically damaged by it all than what I've suggested.