One of the popular characters paired with Sarutobi is Jiraiya. They are both important figures in the Hidden Leaf and their relationship, whether it's as comrades or as teacher - student in some fanfic interpretations, is always interesting to explore.
Obviously Naruto would be one of the main characters interacting with him. Naruto would look up to Sarutobi as a mentor and a source of knowledge. Their interactions could range from training sessions to deep conversations about the future of the village.
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.
Asuma's training might make Naruto more disciplined. Naruto is known for his impulsive nature, but under Asuma's guidance, he could learn to think before acting.
In a Naruto saves Asuma fanfiction, Naruto could use his new jutsu skills. Maybe he discovers a way to reverse the damage done to Asuma. It could be related to his Sage Mode powers, which he could use to heal Asuma's wounds.
It could be that Naruto has a different view of leadership than Sarutobi. Naruto might see Sarutobi's leadership as being too passive or too focused on maintaining the status quo. He, with his more energetic and revolutionary ideas, may resent Sarutobi for not being more progressive in making the village a better place for everyone, including himself.
One characteristic could be the exploration of relationships. Since it's a fanfic, it might delve into relationships that Sarutobi has with other characters in the original work or even create new relationships. The 'lemon' aspect, if not referring to something inappropriate, could imply a certain zesty or interesting element in the story, like a plot twist or a unique interaction that adds flavor to the overall narrative.
In this fanfiction scenario, Naruto might learn unique chakra control techniques from Sarutobi. Sarutobi, being an experienced ninja, could teach Naruto how to better manage his large chakra reserves. This would lead to Naruto being able to perform jutsus more efficiently and with less waste of chakra.
I really like 'Sarutobi's Unseen Trials'. In this fanfiction, it focuses on the political aspects that Sarutobi had to deal with in the Hidden Leaf Village. It gives a new perspective on his leadership and the difficult decisions he had to make.