He probably tries to stay calm at first. Percy has been in tough situations before, so he would assess the situation.
At the start, Percy could be shocked that he's been captured by Artemis. But then his survival instincts kick in. He might start bargaining with Artemis. He could offer to help her in some way in exchange for his freedom. He would also be thinking about his friends and how they could come to his rescue. Percy would likely be observing everything around him, from the behavior of Artemis's hunters to the layout of the place. And he'd be constantly formulating plans in his head to either escape or turn the situation around so that he's in control.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
Percy would probably be really confused at first. I mean, one moment he's a normal demigod and the next he's a jackalope. He'd likely try to figure out what happened and how to change back right away. He might be a bit frustrated with Artemis for doing this to him without much warning.
He may also try to figure out what went wrong. Percy is a smart demigod, and he would likely not just accept the betrayal without trying to understand the cause. So, he might start investigating, talking to other characters who might have some insight, like the other demigods or even some minor gods who are friendly to him.
Percy might be devastated. He could become withdrawn and moody, shutting out his friends and allies. He might also be filled with anger and a sense of betrayal, which could lead him to make rash decisions.
Artemis would be in shock initially. Given her status as a powerful and independent goddess, the idea of being forced into marriage is unthinkable. She would likely try to use her divine powers to break free from the situation. If she can't do it immediately, she might start to negotiate. She could offer other things in exchange for her freedom from this marriage. And if Percy shows any sign of taking advantage of the situation, she would not hesitate to put him in his place.
Initially, Percy is likely to be heartbroken. He thought he had a certain relationship with Artemis, whether it was friendship or respect. So when she betrays him for Orion, he may go into a state of self - doubt. He could wonder what he did wrong and start to isolate himself. But as time passes, he may gather his strength and decide to face Artemis and Orion head - on, to either regain Artemis's trust or show her the error of her ways.
He might be confused at first. Percy has always been a bit of an outsider among the gods, but he may not expect Artemis to hate him. So he would try to figure out what he had done wrong.
Grover would be really sad and sympathetic. He'd stay by Percy's side constantly, using his nature - related powers to help Percy get around. For example, he could use his ability to communicate with plants to create a sort of 'guide' for Percy in the forest or other natural areas. He'd also try to cheer Percy up with his music and his gentle nature.
She could also be angry. Given her power and status, she might feel slighted by Percy. She may take some form of revenge, not in a malicious way but perhaps by making Percy's life a bit more difficult during his quests. For example, she could send her hunters to interfere or withhold some information that could be useful to him.
She would probably be furious. Artemis has a strong sense of justice and protecting the vulnerable. So, she would likely immediately start to plan how to deal with those responsible for Percy's abuse.