One possible reason could be Naruto's natural sense of justice. He can't stand to see a child in danger, and Koyuki being a child at that time, Naruto's heroic nature kicks in. He has always been the type to help those in need.
There could be a fanfic where Naruto has to save a cheerleader in his world, just like the concept from 'Heroes'. Maybe the cheerleader has some special power or is a key to preventing a great disaster in the Hidden Leaf Village. Naruto, with his never - give - up attitude, would surely rise to the occasion.
Well, in most cases, Naruto might use his signature jutsu like the Rasengan. He could either deflect the fire ball with it or use his incredible speed to get to Ami before the fire ball hits her. Maybe he also uses his shadow clones to create a distraction for the attacker while he rescues Ami.
Well, in many fanfictions, Naruto might use his signature jutsus like the Rasengan. He could sense the danger the woman is in using his enhanced ninja senses and then quickly dash in to counter Rizevim's attack with his powerful moves, overpowering Rizevim and rescuing the woman.
Well, Naruto is known for his quick thinking. So perhaps he quickly assesses the situation and uses the surrounding environment to his advantage. For example, if there's a water source nearby, he could use his water jutsu skills (if he has them in that fanfiction) to counter the fireball and save Ami. Or he might even use his body as a shield if it comes down to it, relying on his Uzumaki durability.
One way is to make the cheerleader a vessel for some powerful and positive energy. If she is killed, this energy will be released in a negative way, causing destruction. Naruto, with his strong sense of justice, would sense this and do everything to protect her.
Sakura would be a likely candidate. As a skilled medical ninja, she has the knowledge and ability to diagnose the poison and attempt to create an antidote.
Sakura could be the one to save Naruto. She is a skilled medical ninja. She could use her medical jutsu to keep Naruto alive even while he is crushed. Then, with the help of others like Kakashi, they could figure out a way to remove the rock and fully heal Naruto.
Sasuke could be the one to save Naruto. Despite their past differences, Sasuke still has a connection to Naruto. He might sense Naruto's danger and rush to his aid, using his Sharingan to find a way to safely get Naruto back up from the bottom of the cliff.
In some fanfictions, Jiraiya might save Naruto by teaching him powerful new jutsus. For example, he could pass on some secret toad - related techniques that Naruto uses to defeat a powerful enemy.