Perhaps there could be a villain in the crossover fanfiction that is a threat to both worlds. Saitama and Ainz have to team up to defeat it. Saitama is used to soloing enemies, and Ainz is used to having his minions do the work. So, they have to overcome their differences in fighting styles and learn to work together. Ainz might be in awe of Saitama's sheer strength, and Saitama might be a bit weirded out by the undead nature of Ainz and his minions.
One idea could be that Saitama from One Punch Man somehow ends up in the world of Overlord. He might be mistaken for a powerful player at first. Since he's so strong, he could easily defeat many of the powerful monsters in the Overlord world, which would cause a stir among the various guilds and NPCs. Another plot could involve the characters from Overlord trying to understand Saitama's power, which is beyond their comprehension as it's not based on magic or typical RPG stats.
Another plot could involve a crossover with a different anime universe through a strange portal. For example, if he entered the world of My Hero Academia. Garou could initially try to assert his dominance but quickly realizes that the quirks in that world are different from his martial arts skills. He might then start to learn about these new powers and how to adapt, all while causing chaos and also forming unexpected alliances.
A crossover with 'Dragon Ball' would be cool. Picture Saitama facing off against Goku. It would be an interesting contrast between Goku's love for a good fight and Saitama's boredom with opponents who can't challenge him.
How about a crossover with 'Attack on Titan'? Saitama could find himself in the world of the Titans. His ability to defeat any enemy with one punch would be extremely useful against the Titans. He might not understand the complex politics and fear that surrounds the people in that world at first. But as he starts taking down Titans left and right, he would become a sort of legend among the survivors.
One idea could be a crossover with Dragon Ball. Saitama could participate in the World Martial Arts Tournament and effortlessly defeat all the opponents, creating a hilarious situation. His nonchalant attitude would contrast sharply with the intense and competitive nature of the Dragon Ball characters.
The plot could involve Saitama from One Punch Man being transported to the world of Overlord. He might initially be confused by the different magic - based setting. Since Saitama is so overpowered, he could end up accidentally causing chaos in the new world. For example, he might punch an enemy so hard that it creates a massive crater, which the inhabitants of the Overlord world have never seen before.
One idea could be that Naruto and Saitama somehow end up in a shared dimension where they have to team up to defeat a common enemy. For example, a powerful demon from Naruto's world that has the ability to regenerate like some of the monsters in One Punch Man's world. Another plot could involve Naruto learning Saitama's training regime and trying to incorporate it into his own ninja training, while Saitama is intrigued by Naruto's chakra system and tries to figure out if he can use it to make his punches even more powerful.
One plot could be Tornado having to team up with a new, unexpected hero to face a powerful villain that even she has trouble with. This would show a different side of her as she has to work with someone she might not initially get along with.
A plot could involve a villain who has the power to control hair and targets Saitama because his hair is so unique. This leads to a battle where Saitama has to figure out how to fight without using his hair as a weapon, which is quite difficult since his every move usually involves his head (and thus his hair) in some way.
One idea could be Issei getting transported to the One Punch Man universe and trying to prove his worth among the powerful heroes there. He might start by challenging some of the lower - ranked heroes and learning the ropes of this new world.