Perhaps the Akatsuki saves Naruto because they want to use him for their own purposes in a different way. Instead of killing him or using him as a weapon against Konoha in the traditional sense, they might save him from a situation where he is being hunted by other forces. Then they plan to train him or manipulate him in a new way to further their own goals, like changing the power structure of the ninja world.
Obviously, Naruto is one of the main characters. He is the one being saved. And then, members of the Akatsuki like Itachi, Kisame, or Pain would likely be main characters as well. Itachi, for instance, has his own complex motives and backstory that could play into the story of saving Naruto. Kisame is a powerful and loyal member of the Akatsuki who could be involved in the rescue mission. Pain, as the leader in some cases, would also have a significant role in deciding whether or not to save Naruto and how to go about it.
Well, Naruto is known for his determination. He would probably first try to understand Yagura's situation. Maybe he discovers that Yagura was being manipulated by the Akatsuki through some sort of jutsu. Naruto could then use his own chakra to counter that jutsu and free Yagura's mind. After that, Yagura might be so grateful that he helps Naruto in his fight against the Akatsuki.
Naruto might approach Yagura with honesty. He would probably start by telling Yagura that he knows he's in a bad situation with the Akatsuki. Naruto could share his own story of being ostracized in Konoha at first and how he overcame it. This would make Yagura see that Naruto can understand his pain and might be more receptive to Naruto's help.
Shikamaru might be involved. He's very intelligent and could come up with a clever plan to outmaneuver the Akatsuki. His shadow - binding jutsu could also be used to immobilize some of the Akatsuki members.
Well, Naruto is known for his determination. So in this 'Naruto saves Fuu from Akatsuki' fanfiction, he would likely draw on his experiences and the bonds he has with his friends. He might first try to negotiate with the Akatsuki, but when that fails, he'll engage in a fierce battle. He'll use his speed and agility, which he has developed over years of training, to dodge the attacks of the Akatsuki and rescue Fuu. And Fuu, on her part, might also have some unique abilities that could assist Naruto in their escape from the Akatsuki.
Well, Naruto has a strong sense of justice. In the 'Naruto Saves Yugito from Akatsuki Fanfiction', he would probably draw on his experiences with other Akatsuki encounters. He might call upon the power of the Nine - Tails if necessary. Also, his bond with his friends could give him the extra motivation and strength to face the Akatsuki and save Yugito. He could team up with some of his comrades who also oppose the Akatsuki's actions.
Well, a plot twist might be that Naruto actually gets captured by the Akatsuki while trying to save Gaara. Then, Gaara, in his weakened state, has to find a way to save Naruto instead. This would completely change the dynamic of the story. Or, it could be that the Akatsuki's real target isn't Gaara at all, but Naruto, and they use Gaara as bait. So when Naruto arrives to save Gaara, he walks right into their trap, but then uses his wits and new jutsu to turn the tables on the Akatsuki.
Well, in many such fanfictions, Gaara might use his sand powers. He could sense Naruto is in danger through his connection to the sand or his own instincts. Then, he'd quickly move in, using his sand to shield Naruto from any attacks by the Akatsuki members, like blocking their jutsu with his powerful sand barriers.
Naruto's character could develop in terms of his strategic thinking. Since saving Gaara from the Akatsuki is no easy feat, he might have to come up with more complex battle plans. This would show his growth from a brash fighter to a more calculating ninja.