He might meet Hinata. She is kind - hearted and would surely be interested in the puppy. She could offer to help Naruto take care of it, like making a little bed for the puppy.
In the fanfiction, Naruto might take the puppy back to the village. He could name it something cute like 'Kiba Jr.' and train it to be a helpful ninja dog.
Naruto might run into a group of rogue ninjas who are using the bunker as their hideout. This would lead to a confrontation where Naruto has to use his skills to defeat them and protect the village from their potential threats.
He could meet the previous Hokages. Since the library is infinite, there might be records or even some kind of spiritual manifestation of them. They could guide Naruto further in his journey.
Naruto could encounter a group of rogue ninja who are also after the treasures in the Uzumaki temple. Then he'd have to fight them off to protect the temple and its secrets.
He might meet an old hermit who has been guarding the temple for centuries. The hermit could have valuable knowledge about Naruto's parents or some secret ninja arts.
He might meet the spirit of an ancient ninja who was the keeper of the library. This spirit could guide Naruto through the library and teach him things that no one else in the present day knows.
He might meet an ancient sage who has been sleeping in the ruins for centuries. This sage could have a vast amount of knowledge about chakra and the ninja arts that Naruto could learn from.
He might meet the spirit of an ancient ninja.
Sasuke could also be a main character in some Naruto puppy fanfiction. He might have an interesting relationship with the puppy. For example, at first he might be dismissive of the puppy, but as the story progresses, he realizes the puppy's importance. Another possible main character could be Sakura. She could be the one who helps Naruto take care of the puppy, especially if it gets injured or sick.
He might meet other ninjas who have gone rogue or are in hiding for various reasons. These could be ninjas who have had a falling out with their clans or the village leadership.