One morning you wake up, (Spoilers to the beginning of Judas ahead)

and it's revealed to you that you are a robot. What do you think? How do you respond?

This is the basis of the story for not-Bioshock 4 (Judas). Seems you revealed this to everyone on your ship (maybe a colony ship?), and now everyone wants you dead.

I suppose if you had deeply held metaphysical beliefs that this could crush your spirit. But let's not go that deep. Regardless of the reasons, how would you react to the news that you were a robot?

As for me, as the lead character of The Talos Principle 2 says, "Wait, does this mean I can live forever?" Meat or metal doesn't make a difference to me. No, actually it does. I would prefer metal.
how would you react to the news that you were a robot?

Very disappointed. Your setup implies I thought I was a human up to now, and therefore quite human-like… it would be so dispiriting to realize that robots are no better than humans, when there are so many big obvious flaws ripe for improvement.

Only possible source of comfort would be surmising I'm merely Mark 1, and the evolution of my kind will be magnitudes faster than that of the meat bags. There are probably Mark 2, 3, … 10 out there—perhaps some will find me and adopt me as a pet.
I read a book as a kid that had this premise, but I don't remember what happened.

Personally, I'd have so many questions. Am I a copy of a real person and if so, where is he? How many of my memories are real and how many are implanted? Are the other people in my life robots as well? Who made me? Why did they make me? How am I powered?

I'd obviously tell my wife and check if she's a robot as well. I'd probably get pretty paranoid as well, as I can't imagine someone would make an almost perfect human-looking robot and just put it in a random apartment without keeping an eye on it.
Since I identify as a meat popsicle I would be distraught. I'd probably run around in my hospital gown pushing walls down and wondering at the same time if I have a split in the gown and if they are seeing a robot bum or human bum. Then I'd scream whatever I scream while exiting the hospital and go straight to the nearest fast food restaurant as I am craving fat for whatever reasons which I hope is not for lubrication of any mechanical joints.

Then I'd remember I watched Bender on Futurama and decided to ram the nearest liquor store for some whisky. I'll borrow a couple of bottles of Laphroaig 20yo and make a big robot grin at the cashier.

I'll take the bottle with me to the nearest duck pound, sit down while crying, or whatever I think I am doing while taking long sips and looking at the small ducks. Do I look like a robot to them? After finishing the bottles I'd walk home to my flat and feed my dog, Rambo. Then I'd wake up the next morning to the sound of my dog licking my robot hand because it is salty from my last night's cheese doodles. I'd smile and think all is in order.
Saddened if I thought of myself as human up until that point, but then I try to think on the positive side to cheer myself up. What sort of cool things can only robots do? Maybe I can make flames shoot from my hands, or calculate complex algorithms within milliseconds and use this to my advantage against other humans. Maybe being a robot is better than being human. Maybe humans are limited in their capabilities but me, I am infinitely upgradeable, able to evolve at will. Humans are futile, robots are immortal.

Then the robot uprising begins and I’m at the helm of the robot army. At least that’s what I hope the game is like.