Tsuki wa Higashi ni Hi wa Nishi ni: Operation Sanctuary(2004)

Naoki Kuzumi, a junior in high school, had lost his parents in an accident five years ago, and is living with his aunt, uncle, and cousin Matsuri. He always thought his life was ordinary, besides the fact that he can't remember what happened in his youth. One sunny day as he is taking a nap on a bench, a redheaded girl -- Mikoto -- literally falls on him from the sky. For some reason, she thinks he's her younger brother. Naoki's time period is a temporal shelter for those in the future, when many are suffering from an incurable disease. When she heard her younger brother Yuusuke had been taken to Naoki's time, Mikoto herself had gone back in time.

(Source: Anime News Network)