Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
Yes, Batman Begins can be considered a mystery fiction to some extent. It has elements of mystery as Bruce Wayne uncovers secrets and faces hidden threats in Gotham City.
Naruto could find some similarities between his own values and Batman's. Both are determined to protect their respective worlds. However, the storytelling style in Batman fanfiction might be a new concept for Naruto. He may enjoy the complex characters and the dark, gritty atmosphere often found in Batman stories, or he could be put off by it as it's so different from the more straightforward good - vs - evil in his own world.
One interesting 'batman reaction fanfiction' story could be about Batman's reaction to a new super - villain in Gotham. The villain might have a unique power that challenges Batman in ways he hasn't been before. Another could be his reaction to a new Robin joining the team. Batman has to adjust to training a new sidekick and protecting them while they learn the ropes.
One idea could be that the OC is a childhood friend of Bruce's who has a unique skill, like being an expert in ancient martial arts from a far - off land. They could influence Bruce's training and view on justice in a different way. For example, they might introduce him to more spiritual aspects of combat and self - discovery.
Batman Begins tells the origin story of Bruce Wayne. He witnessed the murder of his parents as a child, which deeply traumatized him. This event led him on a journey to fight crime. He trained with the League of Shadows, learning various combat and stealth skills. But he ultimately rejected their extreme methods. Returning to Gotham, he used his family's wealth and resources to create his Batman persona, with the intention of cleaning up the crime - ridden city.