At first, Percy could be in denial. He might think that there's some misunderstanding, that Annabeth doesn't really mean to break his heart. So he tries to talk to her, to make things right. But as the truth sinks in, he could be really depressed. He might lose his appetite, not be able to sleep well, and just generally seem like a shell of his former self. His whole world revolved around Annabeth, and now that it's falling apart, he doesn't know how to cope.
He could also become angry. Percy has a bit of a temper, and he might lash out at Annabeth, demanding an explanation. He might feel betrayed, especially if he doesn't understand the real reasons behind the break - up. In his anger, he could distance himself from their mutual friends and go on solo quests to clear his head.
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 He could become withdrawn and moody, shutting out his friends and allies.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.
He could also become withdrawn and depressed. Percy and Annabeth have a very strong bond, and her death would leave a huge void in his life. He might isolate himself from his friends and family, lost in his grief and memories of her.
On the other hand, she could try to honor Percy's sacrifice. Annabeth might focus on the reasons why Percy died for her and strive to fulfill any tasks or goals that he might have hoped she would achieve. She would carry on his legacy in a positive way, even though she is heartbroken.
Annabeth could also try to solve the problem that's making Percy cry. If it's related to a quest or a danger, she'll start thinking of ways to fix it. In some fanfictions, she might say 'Don't worry, we'll figure this out together.' and then start making a plan.
He could become withdrawn. Percy might start to distance himself from Annabeth, not wanting to be around her if she's going to be cruel. He might go off on his own and think about what she said, trying to figure out if he did something wrong.
Annabeth could also be angry, not at Percy but at the situation or the forces that led to his death. She might vow revenge and start on a journey to find out who or what was responsible for Percy's sacrifice and make them pay. This could lead her into a whole new set of adventures and challenges as she tries to come to terms with her loss.
She might be heartbroken at first. She could try to jog Percy's memory by showing him things that were important to them.
Well, in many such fanfictions, Percy might be extremely flustered. He could blush furiously and quickly look away, being the gentleman that he is. After all, he has deep respect for Annabeth.
Well, it could be that Annabeth betrays Percy's trust. Maybe she lies to him about something really important, like her true feelings or her actions. This lie could be the start of breaking his heart.