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.
I'm not so sure there's necessarily an 'at' dropped here - 'what time' means 'when' as much as it means 'what hour':
I don't know when he'll come
I don't know what time he'll come
Fine by me. And certainly there's no 'should' involved. All these are fine:
I don't know when he'll come
I don't know what time he'll come
I don't know what time he'll come at
I don't know at what time he'll come
In fact, I reckon the last, avoiding the hanging participle, is the least frequently used of the above, especially in casual speech.
'Is Julie here yet?'
'No, I've no idea at what time she'll come.'
Mmm, I wouldn't say that, but then, that's me.