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.
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.
The emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
Well, Minato might constantly be by Naruto's side in the fanfiction. He could prevent Naruto from taking on any dangerous missions, even if Naruto is capable. For example, he may interfere when Naruto tries to join a risky battle against a powerful enemy, saying it's too dangerous for him.
One reason could be guilt. Since Minato wasn't there for Naruto when he was growing up in the original story, in the fanfiction, he might be overcompensating for that lost time. He feels like he has to make up for all those years he wasn't around to protect his son.
Minato could also teach Naruto about leadership. As the Fourth Hokage, he led Konoha. He could share his experiences on how to make tough decisions for the good of the village and its people. He might tell Naruto stories of past battles and how he strategized to win, which would help Naruto in his future as a leader in Konoha.
Kushina could be an important character even though she's dead in the original story. In this fanfic, her spirit or memories could play a role in Naruto's upbringing. For example, Minato could tell Naruto stories about Kushina's feisty personality and her love for the village.
On a more personal level, they could face internal guilt. As Naruto grows up and achieves great things on his own despite their neglect, Minato and Kushina may realize what they've missed out on. This realization could cause them a great deal of internal turmoil and regret for not being there for their son when he needed them.
In some fanfictions, Minato saves Naruto physically. For example, he might use his teleportation jutsu to appear right in the middle of a fight and whisk Naruto away from danger. In others, he could save Naruto spiritually. He might enter Naruto's mind, like in the fanfic 'Minato's Mind - Saving Mission', and give him the strength and confidence to overcome his inner demons. Sometimes, Minato saves Naruto from a political situation. For instance, in 'Minato and the Hidden Leaf's Politics', he uses his influence and wisdom to protect Naruto from being used as a political pawn.