"Ku sil on aina" is a more colloquial way of saying "kun sillä on aina" which again is an informal way of saying "koska hänellä on aina". "Sillä" refers to this Ämpäri fellow.
I believe that the expression "pää ämpärissä" just implies that Ämpäri gets so drunk that he has to throw up, thus having his head in a bucket. I haven't seen the show so I can't be sure but that's the first thing that came to mind. I don't think there's an idiom involving the phrase.