Yes, in many fanfictions, Naruto is often depicted as a huge fan of the Fourth Hokage. After all, the Fourth Hokage is his father. Naruto grows up idolizing the Fourth Hokage's heroic deeds and wishes to be like him.
In this fanfiction, Naruto might finally get the fatherly guidance he always craved from the Fourth Hokage. Maybe the Fourth Hokage would teach Naruto some long - lost jutsu that only he knew. It could also be a chance for Naruto to understand his father's sacrifices on a deeper level.
In this fanfiction, the plot might involve Naruto being tricked by an evil force into believing that the Fourth Hokage has some dark secrets. So he decides to fight him. It could also be a scenario where Naruto enters a different dimension where the Fourth Hokage is being controlled by an outside power, and Naruto has to free him by fighting. And during the fight, Naruto discovers new things about his father and his own abilities.
One option is a new, original villain created just for this fanfiction. This villain could be someone who wants to rewrite history and prevent the Fourth Hokage from ever becoming Hokage. So they target Naruto as he is the son of the Fourth Hokage. They might have some sort of time - traveling or reality - warping powers.
Naruto himself is often a popular character in these fanfictions. Since he is the main character of the original series and has a connection to the Fourth Hokage as his father, his presence adds depth to the story. Kakashi is also common. His relationship with the Fourth Hokage was that of a student - teacher, and it can be further explored in fanfictions.
It could strengthen Naruto's relationship with Iruka. Since Naruto is the son of the fourth hokage, Iruka might see Naruto as an even more important part of the village and be even more protective. For example, he could be more involved in Naruto's advanced training.
One interesting Naruto Fourth Hokage fanfiction could be one where he survives the fight against the Nine - Tails. In this kind of fanfiction, he gets to see Naruto grow up and train him personally. It would be a heartwarming story of a father - son relationship that was cut short in the original series.
One possible plot twist could be that Naruto has to face an unexpected enemy from the past of the Fourth Hokage. Maybe a hidden group that was against Minato's policies and now they see Naruto as a threat. Another could be that Naruto discovers a new jutsu that Minato was developing but never completed, which changes the power dynamics in the village.
It's really hard to say. The Fourth Hokage was incredibly powerful, with his Flying Thunder God Technique and all. But Naruto has the Nine - Tails' power and has grown a lot. If it was a fair fight, it might depend on who has the better strategy. The Fourth is more experienced in combat, but Naruto has that raw power. So it could go either way.
It could be that in this fanfiction, Naruto has been influenced by some false information about the Fourth Hokage. Some villains might have spread lies about the Fourth Hokage's actions, and Naruto, being young and naive in the story, believes those lies and thus starts to hate the Fourth Hokage.