It could be that he was chasing an enemy or a valuable item that ended up in Hozukijou. Naruto is often on a mission to protect or retrieve something, and this place might be the destination of such a quest.
Another reason could be that he was simply exploring. Naruto has an adventurous spirit, and Hozukijou might be a place that has always intrigued him or that he heard rumors about. So, he decided to check it out, not really knowing what he would find there but being eager to discover something new.
It's possible that there's a strange illness going around the village and Naruto goes to the hospital to see if he can offer any help. He has a strong sense of justice and would want to do whatever he can to stop the spread of the illness and make sure everyone gets better.
In fanfiction, Naruto could be driven to the library by his desire to understand the world outside of Konoha better. He has seen so much in his adventures, but there's still a lot he doesn't know. The library could have books about different lands, their cultures, and their forms of ninjutsu. By studying these, Naruto can become a more well - rounded ninja and gain a broader perspective on the ninja world as a whole.
He might face unique challenges. Hozukijou could have its own set of rules and obstacles that Naruto has to figure out, like perhaps different types of jutsu or magic that are specific to that place.
In some fanfiction concepts, Naruto may go to Orochimaru for the sake of a friend. For instance, if a friend is dying and Orochimaru has the cure or a way to save that friend. Naruto is very loyal and would do almost anything for his friends. So, he might take the dangerous step of going to Orochimaru, even though he knows how dangerous and untrustworthy Orochimaru is.
One reason could be for guidance. Naruto often looks up to Kakashi and might need his advice on a difficult situation, like dealing with a new powerful enemy.
They may be hiding because of a prophecy. There could be a prophecy that says Naruto and Temari together can change the fate of the ninja world in some way. But there are also forces that don't want this to happen. So, they hide to protect themselves until they can figure out how to fulfill the prophecy safely.
They could be on a mission to exchange magical knowledge. Hogwarts has its own set of spells and magical techniques, and so does the American wizarding world. They might go to America to learn new things and share their knowledge as well.
Perhaps he was on a quest for a rare magical ingredient only found in PT.
It might be related to Hermione's research. She may have read about some unique magical phenomenon in Australia that she wants to study first - hand. And Harry, being her loyal friend, goes along with her. For example, there could be a special type of plant with healing properties that only grows in certain parts of Australia.
They might go into hiding because there are still some remnants of Voldemort's followers who are seeking revenge on Harry. These followers could be more dangerous than before as they are desperate and hiding in the shadows. Ginny, being Harry's wife, is also in danger, so they choose to go into hiding to avoid any potential attacks.