I would argue that the first challenge Percy faces in chapter two is an internal one, because his classmates do not believe that Mrs. Dodds was real, and since Percy can remember their former teacher so clearly, it has him questioning his sanity.

To make matters worse, his grades start slipping, and his mother soon receives a letter informing them of his expulsion from Yancy Academy. This challenge doesn't seem to particularly bother Percy, however, and the only final he is interested in doing well in is Latin.

Far more challenging to Percy is the conversation he overhears between Mr. Brunner and his friend Grover, during which Mr. Brunner tells Grover that all they need to do is keep Percy alive until the fall.

Mr. Brunner is the only teacher who Percy respects, and when he calls Percy aside after the exam to tell him that he agrees that Percy should leave Yancy, another challenge is posed, because Percy must overcome the deep sense of shame and embarrassment that he feels.

Once the bus that Percy and Grover take arrives in Manhattan, Percy faces the challenge of accepting a new reality when Grover finally admits that it is his job to protect Percy. Everything is changing and he must rise to the challenge of accepting a new set of circumstances.

At the beginning of chapter 2, Percy struggles with his perception of reality and feels like he is constantly hallucinating. Percy's version of reality differs from that of his peers, who do not remember Mrs. Dobbs being their teacher and believe that Mrs. Kerr has taught them since December.

In addition to his seemingly skewed perception of reality, Percy also experiences trust issues. His best friend, Grover, refuses to confirm his beliefs, and Percy can sense that he is lying to him. Percy's grades also begin to slip and he gets into more altercations with Nancy Bobofit and her friends. To make matters worse, Percy receives an expulsion letter from Yancy Academy.

Percy then overhears his favorite teacher, Mr. Brunner, speaking to Grover about him being in grave danger and avoiding the Kindly Ones. Percy also sees the shadow of a centaur and is even more confused about the identity of Mr. Brunner. During Percy's bus ride home, he confronts Grover about lying to him and is perplexed by the Fates he sees knitting at a nearby fruit stand when the bus breaks down. Percy fears that he is in danger, is upset that Grover is lying to him, and continually questions his perception of reality. At this point in the story, Percy does not realize that he is the son of Poseidon and must embark on an epic quest to stop a devastating war between the gods.

As chapter two begins, Percy's main challenge appears to be the disintegration of his reality. He begins to question his sanity because no one else seems to know who Mrs. Dodds is.

Meanwhile, Mrs. Dodds seems to have been replaced by a Mrs. Kerr, whom Percy has never seen before. To make matters worse, Grover is not able to hide the fact that he remembers Mrs. Dodds and her attack on Percy in chapter one. Percy realizes that everyone is lying to him. At night, he has nightmares about Mrs. Dodds sporting "talons and leathery wings."

Because of his stress, Percy's grades suffer. Essentially, Percy's reality has been disrupted by supernatural events beyond his control. He feels alone, unsure, and frightened. Above all, he misses his mother but knows that he must try to finish out his term at school.

Later, however, Mr. Brunner tells Percy that he must leave Yancy. This adds to Percy's distress. His only comfort is that Grover is taking the same bus into Manhattan. The chapter ends with Percy facing an unknown threat from three old ladies who are knitting a giant pair of socks.

So, Percy's main challenges in chapter two arise from the supernatural events that have contributed to the disintegration of his reality.

In chapter two, Percy faces a rather mundane challenge: school.  First, he learns that he is not going to be invited back to Yancy Academy as a result of his failing grades and increased fighting.  He also has to face exams, though he is especially worried about his Latin exam because he doesn't want to disappoint his teacher, Mr. Brunner, who has always seemed to believe in him when few others have. 

Percy also faces the potential danger posed to him by seeing the three knitting women at the fruit stand on the side of the road -- women we later learn, for sure, to be the three fates.  Further, he witnessed one woman snipping the thread with scissors as they all look directly at him, and Grover tells him that this always means that the observer is about to die.