I'm not sure if there is a specific fanfic with that exact premise. However, in the Naruto canon, Naruto has no such relation to Tsunade. Fanfic writers might create such a story for the sake of exploring different character dynamics and plot twists. They could potentially rewrite the backstories to make it work, like having Naruto be from a different family lineage that is related to Tsunade's in some way.
One possible plot could be that Naruto grows up knowing about his relation to Tsunade. He could be trained in the art of medical ninjutsu by Tsunade's family, and then use his unique Uzumaki chakra to enhance it. As Tsunade's older brother, he might be very protective of her, especially after the death of her loved ones. He could play a key role in helping Tsunade become the Hokage, perhaps by convincing the village elders that she is the right choice, using his own influence as a respected ninja.
Responsibility is a common theme. As an older brother, Naruto has to take care of others. Another theme is protection. He needs to protect his siblings from various threats in the Naruto world.
In 'older brother naruto fanfic', a prevalent theme is family bonds. It shows how the presence of an older brother can strengthen Naruto's sense of family, which he lacked growing up. This often leads to Naruto becoming more confident and having a better understanding of his place in the world. Also, the theme of self - discovery is common. Through his interactions with his older brother, Naruto discovers new aspects of himself, his powers, and his potential. It's like the older brother acts as a catalyst for Naruto's personal growth. Additionally, themes of loyalty and sacrifice are also seen, as the older brother is willing to do anything for Naruto.
One common theme is protection. The older brother is often depicted as protecting Naruto from various threats. Another theme is mentorship, where the older brother guides Naruto in his ninja training.
In this fanfic, Naruto could be a protector for Grayfia. Since he is her older brother, he would feel responsible for her safety. Naruto's ability to sense danger and his combat skills would be useful in Grayfia's often - dangerous world. Also, their different upbringings could lead to interesting cultural exchanges. Naruto, from a ninja village, and Grayfia, from a magic - based society, could learn a lot from each other's lifestyles and traditions.
One popular 'naruto older brother fanfic' is 'The Older Brother's Influence'. It shows how an older brother figure in Naruto's life could have changed his training and relationships. Another one might be 'Brotherly Bonds in Konoha' which delves into the unique bond an older brother could have with Naruto within the village.
Naruto could be developed by having him face moral dilemmas that Tsunade has already experienced. Tsunade can guide him through these, and in the process, Naruto becomes more mature and worldly. Tsunade's character can be developed by Naruto challenging her traditional ways. For example, Naruto might suggest a new approach to handling village finances or medical training, which forces Tsunade to adapt and grow. This interaction would make both characters more three - dimensional in the fanfic.
One possible interesting storyline could be Tsunade teaching Naruto some unique medical ninjutsu skills. This would not only expand Naruto's abilities but also create a strong bond between them as mentor and student.
The older brother in the fanfiction might be a character with a tragic past. For example, he could have been experimented on by Orochimaru and gained some strange powers as a result. And he returns to the village later to protect Naruto from the threats that are coming towards him.