Billy says he came unstuck in time long before he was kidnapped by the Tralfamadorians.
During World War II, Billy is a chaplain's assistant. He's not really supposed to take part in combat missions or anything heroic; he's just there to preach faith in Jesus and play hymns on a portable organ.
Because Billy is totally powerless to do anything important in the war, the men of his unit despise him. He's totally lonely.