Well, he might be really confused at first. Middle Earth has its own rules, magic, and races. Percy, with his water powers, could have some interesting interactions with the rivers and seas in Middle Earth. Maybe he'd help the Fellowship in some way, like using his powers to create distractions during battles or to cross dangerous water areas.
He might meet a hobbit first. Hobbits are curious and friendly, and they live in areas that are likely to be where Percy would first arrive. Maybe he'd stumble upon Bilbo or Frodo's place. The hobbits would be really surprised by Percy, but they'd also be very hospitable.
Percy going to Ilvermorny would be a wild ride. First, he'd be amazed by the castle itself. The four houses would be a new concept for him. He might be sorted into a house based on his heroic nature, like Thunderbird perhaps. He'd introduce the idea of Greek gods to the students there, which would cause quite a stir. And his relationship with Annabeth would be put to the test as they are separated by different magical institutions.
He might meet some talented witches and wizards. Maybe there would be a student who is an expert in magical beasts and they could share stories about their encounters with strange creatures.
Percy might meet an old enemy who has somehow been resurrected or changed in the future. This enemy could be more powerful and have a new plan to take over the world or destroy the demigods. Or he could meet a group of time - traveling demigods from an even further future who are trying to fix something in Percy's future that has gone wrong.
One possible plot could be that Naruto is mistaken for a powerful wizard when he arrives in Middle - earth. He then has to prove himself to the various races there. Along the way, he uses his rasengan to blast through orc hordes and his shadow clone jutsu to create distractions during battles. He also learns about the magic of Middle - earth and tries to combine it with his own ninja powers.
Percy might run into Loki. Loki is known for his trickery. He could either befriend Percy for his own devious plans or try to deceive him. Percy, being used to dealing with tricky situations in his own world, would have to be on his guard against Loki's schemes.
Some minor gods related to the earth like Demeter could be allies. She has her own powers over the earth and could teach Percy some of her techniques. Also, the nymphs, such as the water nymphs and the tree nymphs, could be allies. The water nymphs could help with the water sources within the earth (like underground rivers) and the tree nymphs could help Percy take care of the forests. And let's not forget the pegasi. They could be used for transportation when Percy needs to quickly travel across his earthly domain.