my main verse based on episodes 1, 2 & 3:
Winterfell — Robin is trying to prove himself as a capable and proper Lord. Considering how fragile he was and what was done to him it’s reasonable that everyone wants him to stay at the vale. He’s alone. The only other people he knows and trusts (Sansa and Lord Royce) are at Winterfell. Even with the dead approaching he’s only going to FEEL safe being NEAR THEM. So I don’t see him staying at the Vale or at the very least he would go to Winterfell and try/argue to stay there. He would have arrived before Jon and Dany went to Winterfell.
He’d probably send more fighting men to Winterfell and accompany them as they march North.
Hunting — It relaxes Robin. Much like knitting would for some people. He can concentrate and be intricate with his actions and everything has a cause and affect and he finds that soothing. If he’s heading North it’s likely they’d have to stop and hunt to keep him at ease and it also benefits for bringing more supplies to Winterfell.
Learning — Robin would either be by Lord Royce or Sansa’s side and most likely they would be guiding and tutoring him as they handled all their leadership affairs.
Tyrion — Robin is obviously uncomfortable around Tyrion and will usually keep himself at a distance and not look at him. Mostly because he tends to trigger memories of his mother he’d rather not recall.
Jaime’s Trial — It amuses me to think he’d be sitting with the rest of his family and making some off-hand comment about if they were in the Vale they could throw him through the moon door. Part of it would be a joke but considering things I feel like some people would be all shifty eyes at him. >__>
Independence — Robin considers the Vale and the Riverlands to be part of The North now. The Riverlands since they declared for Robb and The Vale since he declared for Sansa and Jon. He supports their independence.
The crypts — If he stays in Winterfell (most likely due to it being too late for him to leave) he’d be placed in the Crypts. Lord Royce probably arms him with a bow and dragon glass arrows and advises him to protect those he’s in the crypts with as a means persuading him. Robin is also not stupid, he can’t fight and knows he’s a liability on the field.
Final hours — He’s probably with Royce who’s trying to subtly sprinkle in some last minute wisdom onto Robin in case he doesn’t make it. He’s also probably telling Robin stories about Jon Arryn and it’s probably the first time he’s hearing anything genuine about his father and able to retain it.
Fight for the Dawn — Robin is in the crypts while the battle is going on. When the dead in the crypt stir he uses his bow and dragon glass arrows until they run out. Towards the end he has a seizure and is ushered to the very back of where everyone is hiding until the night King is taken out.
The Dawn — As the survivors emerge he realizes Yohn Royce is nowhere to be found among the survivors. This BREAKS him and he cries while he is urged to relax or risk another seizure. It’s the first time since it was discovered he was poisoned that he accepts a low dose of sweetsleep for the nerves. THIS IS SUBJECT TO CHANGE but based on the fact that I don’t see Royce in the promo and he was at the front and then nowhere to be scene at the rest of the battle I assume he is dead.