Izuku would probably be terrified at first. After all, he's been kidnapped by villains. But then, his hero - like determination would kick in. He might start looking for ways to escape or at least resist their plans.
Maybe it's a group of villains working together. They could be from different factions but united by the idea that Izuku has the potential to become a huge threat to their plans. They might kidnap him to either eliminate him or turn him to their side. Also, it could be a lone, obsessed fan of a villain who wants to prove themselves by bringing Izuku to their idol.
All Might could also be a potential hero to save Izuku. He has a strong bond with Izuku and his presence alone could intimidate Overhaul. He would use his experience and power to rescue Izuku from the dangerous situation.
All Might would be furious. He would see it as his failure to protect Izuku. He would probably mobilize all the resources he has, both in his hero form and his connections in the hero world, to rescue Izuku. And he would be blaming himself all the while for not being there when Izuku was kidnapped.
One idea could be that Izuku is kidnapped by a group who wants to use his knowledge of hero strategies against the heroes. They hold him hostage in an abandoned building. Another could be that a villain with a grudge against All Might kidnaps Izuku as a way to lure All Might into a trap. Maybe a group of anti - quirk fanatics take Izuku because they believe his powerful quirk is a threat to their cause.
One possible plot could be that Izuku's friends discover his kidnapping and start a rescue mission. They might face various obstacles set by Overhaul and his minions while trying to find Izuku's location.
Well, in a lot of such fanfictions, Izuku might be forced to join the League of Villains against his will. They could try to brainwash him, using their various quirks and methods. Maybe they'd want to use his knowledge of hero strategies against the heroes.
One common plot twist is that the kidnappers turn out to be working for a more powerful villain who has a personal vendetta against Izuku. Another could be that Izuku's friends think he's been kidnapped by one group, but it's actually another. For example, they might suspect a rival hero school, but it's a secret underground organization.
He would likely be more confident. Having the powers of the Human Torch gives him a new level of strength. So he might stand up straighter and be more assertive in his actions.
In many fanfictions, Izuku adapts by relying on his heroic spirit. He sees the similarities between his world and the Spider - Verse in terms of helping people. So he starts to work with the local heroes, like Peter Parker or Gwen Stacy. He learns from them about the different threats in the Spider - Verse and how to handle them. He also tries to incorporate his own fighting style from My Hero Academia with his new spider - based powers. For instance, he might combine his One For All power with his new spider - strength for more powerful attacks. This process of adaptation is really exciting to read about as it shows Izuku's growth in a new and different environment.