Maybe a dark version of Neji himself could be the antagonist. This could be a Neji from a parallel world where he succumbed to hatred and envy. He might try to destroy the 'good' Neji and take over his life, believing that his way is the right way and that the other Neji is a weak version.
Another option is that a powerful and unknown entity that exists between the different realities where the Nejis come from. This entity could be threatened by the fact that the two Nejis are interacting and trying to merge their knowledge and powers, so it tries to stop them by any means necessary, including pitting them against each other.
Perhaps Neji's own insecurities could be the main antagonist in a sense. When he meets his alternate self, he might start to feel inadequate or question his own identity. This internal struggle could be the main obstacle they have to overcome rather than an external enemy.
In such a fanfiction, Neji might initially be in shock when he meets his alternate self. He could start to question his own life choices and beliefs. For example, if his alternate self has made different decisions regarding his loyalty to the clan or his relationships with other characters, it would force the original Neji to re - evaluate his own stance. Maybe they would even engage in a sparring match to test each other's skills, which could lead to a new understanding of his own abilities.
Perhaps there is no traditional antagonist. Instead, the conflict could be within Percy himself. His future self might have some dark secrets or regrets that Percy in the present doesn't want to face. So, in a sense, Percy's own denial or fear could be the 'antagonist' in this fanfiction.
Perhaps Nico's own insecurities could act as an antagonist. His future self might be so different and successful that Nico starts to feel inadequate and his self - doubt tries to prevent him from fully interacting with his future self and learning from the encounter. Another idea is that there could be a group or organization in the story that has rules against time - traveling encounters like this, and they try to stop Nico from having this meeting.
In 'neji meets his alternat self fanfiction', one possibility is that they start to fight at first due to confusion and a sense of threat. However, as they get to know each other, they realize that they have more in common than they thought. They could end up learning from each other, like sharing different jutsu techniques or ways of thinking about their ninja duties.
It could be an evil ninja who wants to stop Kakashi from meeting his mother. This ninja might be from a rival village and fears that Kakashi will gain some powerful knowledge or ability from his mother that could be a threat to their plans.
The main antagonist might be a dark sorcerer. This sorcerer has put a spell on Spyro's mother long ago, trapping her. He doesn't want Spyro to find her because he fears that together they will be strong enough to break his power. He uses his minions to try and stop Spyro during his journey to find his mother.
It could be Frieza. Since Frieza has a history of hating the Saiyans, he might see Goku's sister as a new threat and try to eliminate her.
It could be someone who is jealous of their new - found bond. Maybe a rival of Uzumaki who doesn't want him to have such a powerful ally in his sister.
It could be a dark wizard who is trying to prevent Teddy from meeting his father. This wizard might be afraid that Teddy will gain some powerful knowledge or magic from his father that could be used against him.