One possible plot could be Naruto learning unique jutsu from Minato right from the start. Minato could teach Naruto how to use the Flying Thunder God Technique at an early age. This would change Naruto's battles significantly, as he could quickly move around the battlefield and surprise his enemies. He might also learn about Minato's leadership skills and how to handle difficult situations within the village, making him more respected among the villagers much earlier in the story.
The Akatsuki could still be the main antagonists. However, in this fanfiction, they might target Naruto earlier because they know Minato is training him. They could fear that Naruto would become an even greater threat to their plans in the future. Pain, in particular, might see Naruto as a symbol of the Hidden Leaf's strength that needs to be eliminated. Also, there could be an internal antagonist within the Hidden Leaf, someone who is jealous of Naruto's special relationship with Minato and tries to undermine him.
One popular storyline could be the apprentice learning unique jutsu from Naruto. For example, they might learn his Rasengan technique in a more in - depth way, with Naruto guiding them through every step of the process.
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.
First, know the Naruto universe well. Understand the rules of chakra, the different ninja clans, and the existing jutsu. This knowledge will make your story more believable. For example, if your apprentice is from the Uchiha clan, you need to know about their Sharingan abilities.
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.