Well, their brotherly relationship could also be shown in how they deal with the village's problems together. Naruto's strong will and Minato's wisdom could combine. They might face off against a new, powerful enemy that threatens Konoha. Naruto would be all about charging in, while Minato would plan a more strategic approach, and they'd have to learn to work with each other's styles.
It would change a lot. For starters, Naruto might not have grown up as an outcast. With Minato as his brother, he could have had some family support and recognition from the start.
Their relationship would likely become more equal. In the original story, Minato was more of a distant, heroic figure to Naruto. But through time travel, they can interact as peers, sharing knowledge and experiences. This would deepen their bond.
Well, perhaps a plot twist is that when they time travel, they end up switching bodies. Naruto, in Minato's body, has to deal with the responsibilities of being a Hokage in the past, while Minato, in Naruto's body, has to navigate through Naruto's life in the present. This would create a lot of comical and also very tense situations as they both try to adapt and not mess up the timeline.
One popular theme is Minato getting to know Naruto in the afterlife. Since Minato died when Naruto was a baby, fanfictions often explore their relationship in a spiritual realm.
In fanfiction, Naruto and Minato often represent a connection between generations. Minato was a great ninja in his time, and Naruto is following in his footsteps. Fanfictions can explore how Naruto lives up to his father's reputation.
I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide variety of themes and scenarios, some of which might be rather strange or controversial. However, Naruto fanfiction often explores different relationships and alternate storylines, but this particular one seems a bit unorthodox and I haven't personally come across it.
One of the main similarities is their self - sacrifice. Minato sacrificed himself to save the village and his son. Naruto has also shown that he is willing to put himself in harm's way for the greater good. He's always the first to jump into a dangerous situation if it means protecting those he loves. Another aspect is their leadership qualities. Minato was a great leader as Hokage, and Naruto has developed into a great leader as well, inspiring those around him.
You can try websites like Fanfiction.net or Archive of Our Own (AO3). They have a vast collection of Naruto fanfictions. Just search for 'Minato' and 'Naruto' in the search bar, and you'll get a list of relevant stories.
Minato would likely be extremely shocked. Naruto is his son, and the idea of Naruto attacking him would be completely unexpected. He would probably try to talk to Naruto first, to find out what is wrong and why he is being attacked.