Their relationship would likely grow stronger. Naruto would gain more respect for Hiruzen as he sees the amount of work the old Hokage has to do. And Hiruzen would see Naruto's eagerness to help and grow. This could lead to Hiruzen giving Naruto more responsibilities within the village.
Well, Naruto might bring his unique energy to the task. He could start by getting distracted easily, like seeing a cool jutsu diagram on a paper and wanting to try it out right away. But then he'd remember he's there to help Hiruzen. Naruto might come up with some unorthodox ways of organizing the papers, like sorting them by how much 'fun' he thinks the mission described on them would be. And his presence would surely make the usually dull paperwork a bit more lively for Hiruzen.
Perhaps he uses some ninja techniques like a shadow clone jutsu. He could make multiple clones to help him sort through and complete the paperwork quickly.
In certain fanfictions, Naruto could approach the paperwork in a more diplomatic way. He might convince others in the village, like his friends or some of the ninja clans, to help him with the paperwork. By using his charisma and leadership skills, which he has developed over time, he gets everyone to pitch in. And with their combined efforts, they manage to defeat the mountain of paperwork. After all, Naruto is not just a powerful fighter but also a great leader who can inspire people to work together.
Maybe Naruto is put in a situation where he feels that Hiruzen's way of handling things is the best option. For instance, if there is a major crisis in the village that requires a more traditional and experienced approach, Naruto might choose to act like Hiruzen. Also, he could be influenced by the stories he has heard about Hiruzen and wants to fill his shoes in a way.
It depends on the specific fanfiction. Some might have Naruto develop strategies or skills to overcome the paperwork challenge. Others might not focus on this aspect at all.
Maybe Naruto discovers a hidden scroll that has a spell or technique specifically for dealing with paperwork. He then uses this new - found knowledge to make all the paperwork disappear or transform into something more manageable, like a single small scroll that contains all the important information in a condensed form.
Perhaps in this fanfiction, Naruto has some sort of special power or transformation that he unlocks just in time to face Orochimaru. It could be related to his Nine - Tails chakra in a new and unique way. He uses this power to break through Orochimaru's defenses and save Hiruzen. Naruto might also use his knowledge of Orochimaru's weaknesses that he has learned either from Jiraiya or from his own past encounters with Orochimaru's minions.
In a 'naruto adopted by hiruzen fanfiction', the plot could involve Naruto having a closer bond with the older generation of shinobi. Hiruzen would surely introduce Naruto to his trusted comrades. Naruto could learn unique jutsu from them. Also, his growth as a ninja would be different. Instead of seeking attention through pranks, he might focus more on living up to Hiruzen's name. His relationship with Sasuke could also be affected. Maybe they would become rivals in a more positive way, with Naruto having a stronger sense of family and purpose because of Hiruzen's influence.