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.
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.
One reason is their friendship. Naruto was one of the first people to see Gaara as a person, not a monster. So, Gaara would feel indebted to Naruto and be willing to risk his life. Another reason could be for the sake of the ninja world. Naruto is often seen as a key figure in maintaining peace, and Gaara, as a responsible leader, would understand the importance of saving him.
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.
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.
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.
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.