Shinjuku 新宿区 (Shinjuku-ku) is one of the 23 special wards of Tokyo, Japan. It is a major commercial and administrative centre, home to the world's busiest railway and the Tokyo Metropolitan Government Building. Shinjuku has a role in the story of Final Remix and NEO: The World Ends with You.
A New Day
Neku has visions of Shinjuku being disintegrated by a pulse from a large red Reaper skull in the sky. In the empty streets a mysterious girl (revealed to be Tsugumi Matsunae) stumbles through. He realizes after seeing Rhyme erased by the Dissonance Shark that it is a vision of the future. Shinjuku is erased in an Inversion. Joshua and Hanekoma discuss the Inversion: Hanekoma is curious and sees it as the "calm before the storm", but Joshua is apathetic.
Hanekoma's research in the Secret Report Addendums details the chronology: Shinjuku was Inverted either shortly before or during Coco trapping Neku and Beat in her fake Shibuya. Contrary to Joshua's behavior in the epilogue of A New Day, Hanekoma says the Composer is "rightfully" interested in the Inversion, and Hanekoma is determined to find the consequences of the Shinjuku Inversion.
NEO: The World Ends with You
Several characters in NEO: The World Ends with You come from Shinjuku: these are Shiba Miyakaze, Tanzo Kubo, Ayano Kamachi, Kaie Ono, Hishima Sakazuki, and Shoka Sakurane. They are currently part of the Reapers' Game in Shibuya. It is eventually revealed that Kaichi "Susukichi" Susuki and Tsugumi Matsunae are also Reapers from Shinjuku.
Over the course of the story, the circumstances behind Shinjuku's erasure are revealed. The city's Composer, Hazuki Mikagi, wished to "cleanse" the city and recruited Tanzo Kubo--actually an Angel known as the Executor--to carry this out. Kubo used Shiba as a pawn and turned the other Reapers against Shinjuku's Conductor, who ended up erased. After the Inversion, the Shinjuku Reapers found a new home in Shibuya, though Kubo sought to erase it as well and once again manipulated Shiba into this.
After the exorcism of Kubo and prevention of Shibuya's Inversion, Shiba chooses to return to Shinjuku to rebuild it. Tsugumi, Hishima, and Kaie choose to accompany him and aid in the reconstruction.