The closest thing to that description that comes to mind is Death March, but I wouldn't really recommend it.
IseSma has some episodes like that, although they have a distinct home, a large part of Garo: Vanishing Line, some of GATE.
Then Kamisama no Inai Nichiyoubi, where it's not a guy and doesn't fix a whole lot of problems. If you somehow haven't seen Kino no Tabi, there's that, and again Kino rarely fixes problems.