Writing a scene with from a disoriented viewpoint?

Use body language to describe your characters disorientation, (rather than writing out a train of thought).

Show how unsure he feels about his surroundings by the way he moves around in them.

For example when your character wakes up, did he merely 'step out of the chamber' or did he 'place his foot gingerly onto the unfamiliar cold floor'.

I think in your case the thought process would be slow. I would try to recreate the brain's activity after waking from a dream. For me, that comes with a surreal, peaceful quality. The brain feels malleable after sleep, like only half of it is there and the part that contains the ego and the frantic inner commentary is not yet fully awake. That's my waking state.

Go read "Ghost Story" by Peter Straub and Stephen King. It does this disoriented sensation very well.

Does he need to be your POV character for that scene? I'm not suggesting you jump to someone else just for that, but if you alternate POV at all elsewhere, too that might be the solution.

I would set up a contrast between what he expects to happen and what actually does, since it would be drastically different. I don't know the other circumstances, but as Katherine said, reinforce that unfamiliarity and have his imagination fill in the gaps for why waking up is different than it should be. A semi-dreamlike state has precious little in the way of critical thinking skills; a person could be subjected to anything and just roll with it, feeling that it belongs even if he has to subconsciously make up reasons why.

The first chapter of the book I'm reading right now, "Altered Carbon" by Richard K. Morgan featured something along the lines of waking up from cryo sleep. The character had done it multiple times so that took some of the shock away.

I'd say focus on the physical aspects of it, maybe the character is too muddy from the sleep to contemplate meaning very much, so descriptions of the way it feels - i.e. if it's very cold, the strangeness of the handler's touch, smells and sounds that the character isn't used to, maybe eyesight coming into focus - that sort of thing. Then as the character regains focus you could bring him in during the middle of the handler's explanation of what's going on.

Writing close-third is not that much different from writing first when it comes to experiences the way I do it; just the character's name instead of I in front of heard or felt or whatever.

There are quite a few descriptions of something like this is Dan Simmon's Hyperion series and he mostly focuses on confused thoughts and sensations in the body.

My first thought was: Could you skip over all the coming out of cryo-sleep disorientation? Of course, that popped into my head because of all the advice against opening a novel with a waking up scene.

Since he's being carried, it might be something like... Why is the floor over there? There's a severed head floating over my face!! My feet are cold. What if they drop me?