Well, Naruto has always been adaptable. In a Hive fanfic, he'd first try to communicate with the locals. His positive attitude would make it easier for him to get along. He could use his ninjutsu to his advantage, like using his transformation jutsu to blend in or gather information. He'd also be excited to learn new things in the Hive, which would further his adaptation process. His goal - setting nature would make him look for ways to improve the Hive or help those in need there.
He'd probably start by observing. Naruto is curious, so he'd watch how the Asgardians live, their customs, and how they interact. Then he'd start to adapt by using his ninja adaptability skills.
He would need to understand the social hierarchy in Kirigakure. In Konoha, he was used to a certain way of doing things, but in Kirigakure, there could be different rules regarding how ninjas interact with each other and with the village leaders. For example, he might have to show more deference to the Mizukage than he did to the Hokage. Also, he'd have to adapt to the food. Kirigakure is near the sea, so there would be a lot more seafood in his diet.
Naruto adapts by becoming one with the forest. He observes the animals and learns from their behavior. For example, he sees how squirrels store nuts for the winter and starts doing something similar with his food supplies. He also gets used to the sounds of the forest at night, which at first might have been scary but then becomes a normal part of his life. He may even start using the natural elements of the forest like vines to help him in his ninja training.
Naruto would first have to adapt to the different magic system. Since he's used to chakra, understanding the arcane, fel, and nature magic in Warcraft would be a challenge. But he's a quick learner. He might start by observing the mages in the world. Also, he'd need to adapt to the new races. He'd probably be curious about the elves, dwarves, and orcs, and try to make friends with them using his friendly and outgoing nature.
Cloud would likely have a tough time at first. He's used to a very different kind of world. But his strong will and combat experience would help. He'd probably be confused by chakra at first, but once he understood it, he could use it in his own way, like infusing his sword with chakra for more powerful strikes.
Well, Naruto might adapt to Shunpo in the fanfiction by first observing. He's good at that. He'd watch how others use it, break down the movements in his mind. Then he'd practice, using his determination. His chakra control, which he has improved over time, would also play a role. He'd likely infuse his own style into it, not just copy it blindly.
Naruto would probably find the social hierarchy in Westeros quite strange at first. But his friendly and adaptable nature would help. He might start at the bottom, but his skills would quickly earn him respect. He could become a sort of outsider hero, not fully bound by the strict rules of the hierarchy.
Naruto has been through a lot in his own time, so he may use that experience to adapt to the past. He might start by making friends with the younger versions of people he knows. He'll also have to be careful not to reveal too much about the future to avoid disrupting the timeline. He could use his ninja skills to earn respect and find his place in the past world.