She could be a bit nervous too. In case Percy doesn't do as well as he hoped, she might be worried about how he'll feel. But at the same time, she has faith in him, so it's a mix of emotions. She might also be looking forward to seeing Percy in his element, as he's always so confident in the water.
In the fanfiction, perhaps Annabeth would notice some new skills that Percy has developed in his swimming. She could be really proud of him and later on, they might talk about how he improved. She might also compare his swimming style to his fighting style, seeing the similarities in his determination and speed in both activities.
There could be a school janitor who is actually an old demigod who retired from the hero business. He could give Percy some valuable advice based on his own experiences. And of course, Percy will surely meet Annabeth's classmates, some of whom could be friendly and some not so much, depending on how they react to the new boy in school.
Percy could meet the school bullies. And they might pick on him at first, but then be in for a big surprise when he shows his strength. Also, he might encounter some teachers who suspect there's something different about him. Maybe a science teacher who notices his quick reflexes during an experiment or a history teacher who is amazed by his knowledge of ancient mythology.
He might meet normal high school students. There could be the typical jocks, nerds, and popular kids. Maybe he'd encounter a really smart student who is intrigued by Percy's different way of thinking. Or he could meet some mean students who pick on him at first but then learn to respect him.
Grover could be one of Percy's friends to meet Annabeth. He's a satyr and has been with Percy through a lot. When he meets Annabeth, he might be really happy to see another important person in Percy's life.
One possibility is that Percy clashes with the school's jocks or popular kids. Since he's an outsider, they might pick on him at first. But then Percy shows them what he's made of, not with brute strength necessarily but with his quick thinking and his ability to outwit them. He could also get involved in a school competition, using his unique skills to help his team win.
In a fanfiction, they might meet in a dangerous, enchanted forest. Percy has been lost in there for days, and Annabeth has come to find him. As Percy hears the rustling of leaves, he gets ready to fight, but then he sees Annabeth emerge from the bushes. Annabeth quickly assesses the situation and starts leading Percy out of the forest, all while they catch up on what has happened since they last saw each other. There are strange creatures lurking around, but they feel safer now that they are together.
Percy might feel a deep sense of contentment. He and Annabeth have a special bond, and having her trust him enough to sleep in his lap could make him feel warm and happy inside. He may also feel a bit nervous, not wanting to move and disturb her peaceful slumber.
In many fanfictions, when Annabeth's friends meet Percy, there could be a mix of reactions. Some might be initially cautious, especially if they've heard of Percy's dangerous adventures. But often, they end up being impressed by his skills and his loyalty to Annabeth.
One reason could be that Percy often gets into dangerous situations alone to protect others. Annabeth, being his friend or more, would feel bad seeing him constantly at risk.