In this fanfiction, Minato could show his protectiveness in a very direct way. Maybe he has some kind of special connection with Naruto that allows him to sense Naruto's distress from the afterlife. So when Naruto is in a fight or in a dangerous situation, Minato's spirit could manifest itself and directly interfere with the situation to keep Naruto safe. He could also have left behind some sort of guardian entity or power within Naruto that would take over and protect Naruto when he is unable to defend himself.
Well, in the fanfiction, Minato could be protective by influencing Naruto's teachers or the people around him. He might use some kind of spiritual communication to tell them to look after Naruto carefully. For instance, he could appear in Iruka's dreams and ask him to be extra vigilant about Naruto's safety. And also, Minato could have set up some sort of pre - arranged protection plan in the village that is specifically for Naruto, so that whenever Naruto is in trouble, the village mechanisms would automatically be more inclined to protect him.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.