Can we say: "I don't know at what time he'll come"?
Yes, we can. And we probably should! In a statement, we always use "at" when talking about time (He'll come at 6pm), but natives regularly drop the "at".
I don't know what time he'll come.
What time are you coming?
I'm not sure what time she went home, but it was very late!
Do you know what time you're leaving?
As you can see, "at" has been dropped from all of these but they are all perfectly understandable and commonly used.