It could be an elf. Elves are often on the lookout for strangers in their forests. If Percy enters an elven forest, an elf like Haldir might be the first to encounter him. The elf would be cautious at first, but Percy's friendly nature and his own kind of magic might pique the elf's interest. And they could start a conversation about their different worlds.
There's a chance he'd meet a ranger like Aragorn first. Aragorn is always wandering around Middle Earth, protecting it. If Percy shows up in a dangerous area, Aragorn would sense his presence and come to check it out. Aragorn would be a bit wary but also intrigued by Percy's strange appearance and abilities.
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.
He might first meet a human. Azeroth has a lot of human settlements, and it's likely that Percy would stumble upon a human village or town first. The humans would be both curious and cautious about him, especially if they notice his strange powers.
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.
There's also a chance that he'd meet Lydia first. Lydia has her banshee powers, and she might be drawn to Percy because of his otherworldly presence. She could sense something about him that's not of this world, and approach him out of curiosity. Their meeting could be a very intense and mysterious one, with Lydia trying to figure out what exactly Percy is.
He would probably meet Eragon. Eragon is a key figure in Alagaesia, and their encounter would be full of surprises as they come from different worlds with different powers.
If Percy Jackson goes to Valhalla in fanfiction, he could meet a whole bunch of Norse heroes. There's Sigurd, the dragon - slayer. And then there are the Valkyries who choose who gets to go to Valhalla. He might also run into Loki, who is always causing some sort of trouble. Percy would be in for quite an adventure as he tries to figure out who he can trust among these new and often unpredictable characters.
He might meet a new teacher who is secretly a minor god or goddess in disguise.
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.
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.
He could meet some Aboriginal heroes or spirits. They might have their own powers and knowledge that Percy can learn from.
He might meet Luke before he turned bad. Percy could try to change Luke's fate, convince him not to follow Kronos. It would be a great chance to save a friend and prevent a lot of the chaos that happened later.