It's difficult to say without context. I am guessing that 'done' here means 'exhausted/finished'. If you are going to ask about sentences, please provide a little more context in future.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.
Mister MicawberI rather think one 'in' is missing:The running had done him in in the morning. = The running made him very tired in the morning.That is certainly more natural, but I think it's just about possible without using a phrasal verb - after all that running, he was done (in).