The story could start with Naruto suddenly finding himself in the past with Team Minato. At first, Minato and the others might be suspicious of him. But Naruto could prove his worth by showing his combat skills. Since he has experience fighting powerful enemies, he could assist Team Minato in their missions more effectively. His presence could also change the relationships within the team. For instance, he might form a special bond with Kakashi, influencing Kakashi's development in a new way.
Naruto would have a huge impact. His combat skills, like the Rasengan and his ability to use shadow clones effectively, would enhance the team's fighting power. He could also bring a sense of fun and camaraderie that might be different from what the team is used to. Just think about how he'd interact with Obito and Rin, probably making them laugh with his goofy antics.
One interesting plot twist is if they time travel and find that the world they end up in has a different version of the Chakra system. They have to adapt quickly to this new form of energy while also trying to find a way back home. Also, imagine if they time travel and discover that Naruto was supposed to be born in a different village in this alternate timeline. This would completely change the way they view Naruto's destiny and their role in protecting him.
If Minato time - travels, it could change a lot. For instance, if he prevents the Nine - Tails attack, Naruto would grow up in a very different environment. He might not be the underdog he was in the original story. Also, if Minato changes things during the Uchiha clan's situation, Sasuke's story would be completely different. Konoha's political situation would also be altered. There would be less animosity between different clans perhaps.
Time travel can completely change their missions. They might have to redo a mission they already completed in a different way. For example, if they failed a mission in the original timeline, they could go back and try to succeed this time. Or they might have a new mission altogether because of the changes in the timeline that they caused through time travel.
One possible story could be that Team Minato time travels to the future. They would be amazed by the technological advancements. Naruto might be excited to see how the village has grown. They could also learn new jutsu from the future ninjas and take that knowledge back to their own time to strengthen Konoha.
Well, a frequent plot twist is when the time travel affects the relationships within Team Minato. Maybe they go back in time and realize they were wrong about someone. Or they accidentally change something in the past that makes one of their relationships turn sour. Also, they could find out that someone they thought was an enemy in their original time was actually a friend in the past, which is a huge revelation and changes the whole dynamic of the story.
Jiraiya is likely to be a popular character. He was Minato's teacher and has a strong influence on Naruto as well. In a time travel fanfiction, he could provide a lot of wisdom and help in the different situations they encounter. Also, Tsunade could be important. Her leadership skills and medical ninjutsu could be very useful when they are in the past, and she has her own history with both Minato and Naruto's family.
They would be empathetic. Minato, as Naruto's father, would especially feel a deep sense of pain seeing his son struggle. He would want to reach out and comfort Naruto during his lonely times when the village shunned him because of the Nine - Tails.
They would be shocked at first. Minato would be filled with pride and joy seeing his son all grown up. Kakashi would respect Naruto's strength. Obito might be a bit envious of Naruto's popularity but also admire his spirit. Rin would be very friendly towards Naruto.