Naruto might initially be in shock. He has always counted on Kushina's love, and this sudden abandonment would be a huge blow. He could become withdrawn and quiet, not understanding why his mother would do such a thing.
In the fanfiction, it could be that Kushina is pressured by external forces. For example, there might be a powerful group or individual who forces her to abandon Naruto for his sister. This could be part of a larger political or power - play scheme. Also, it's possible that Kushina has a moment of extreme emotional confusion. She might see something in Naruto's sister that she once desired for herself when she was younger, and in a moment of weakness, she decides to focus on the sister instead of Naruto.
One possible reason could be some sort of dark influence or curse that makes her lose her maternal instincts. Maybe a powerful enemy targets her and in order to protect Naruto from further danger related to her, she feels forced to abandon him. Another reason might be that she is made to believe false information about Naruto, such as that he will bring great disaster to the village in the future.
In such a fanfiction scenario, the village of Konoha would be affected too. Minato is a respected figure. His abandonment could cause a scandal. Naruto might not have the same level of protection from the village elders. This could lead to different power dynamics within the village. Maybe other ninja clans would try to take advantage of Naruto's situation. Also, Kushina's family might react in various ways, some might try to find Minato and some might focus on taking care of Kushina and Naruto.
He may develop trust issues. Being abandoned by his mother could make him more cautious about getting close to people. However, this could also make him more determined to prove himself. He might strive to be the best ninja in the village to show that he is worthy of love and acceptance, despite being abandoned. For example, he could push himself extra hard during training.
Communication is key. Naruto and his sister could sit down and have an honest conversation about what's bothering him. They could clear up any misunderstandings that led to his hatred.
Well, it could go in many directions. For example, his family might start to search for him, which could lead to a series of adventures where they encounter new characters and face various challenges. His wife, if he has one in the story, might become stronger and more independent in his absence, vowing to bring him back.
Naruto and Lemon's relationship could start off as a typical sibling one, with some bickering but also deep love. As for Kushina, Naruto would have a great sense of respect and love for his mother. Lemon, on the other hand, might feel a bit left out at first since she didn't get to know Kushina as Naruto did. But as the story progresses, through memories and stories about Kushina, Lemon would develop a strong connection to her too. They would all support each other in battles and in life in general.
One possible plot could be that Lemon, Naruto's sister, discovers some hidden jutsu that Kushina once knew. Naruto and Lemon then train together to master it. Along the way, they face challenges like other ninja clans trying to stop them as they fear the power of this new - found jutsu. And in the end, they use the jutsu to protect their village and honor Kushina's memory.
I'm not sure there is an official 'Sister Lemon' in the Naruto canon. But in a fanfiction, perhaps this 'Sister Lemon' could be an original character. Maybe she could be a long - lost sister who has some unique powers related to the Uzumaki clan, like Kushina. She could play a role in Naruto's growth and his understanding of his family's past.