In reply to ruzuzu: I think some folks have a tendency to duplicate a preposition for verbal padding: I took it off of the shelf; she took the washing in off of the line (an extreme example, but not impossible, which just means "she took in the washing").
In reply to pterodactyl: "When" shall I pick you up is less specific than "(at) what time"; the answer "this afternoon" is more general than "at 3 pm" Therefore the wording of the question is determined by how specific I wish the answer to be.
Prescriptive grammarians would probably say yes, it is an error, because it's missing the word "at". It should be "What time should I pick you up at?" or "At what time should I pick you up?"
If these grammarians are also copy editors, they might suggest the phrasing "When should I pick you up?", which avoids the "what time" construction entirely. After all, English already has a perfectly good word for asking questions about time ("when"), so, the editors might say, why not just use it?
Actually, I think there's a jolly good reason to use "what time" instead of "when". I don't think they mean the same thing.
My idea is that people use "what time" to refer to time on a clock, as opposed to time on a calendar. For example, if you ask someone "When did you arrive in London?", they might answer "Last Thursday", which isn't helpful if what you're really inquiring about is the arrival time of their train. So, instead, you can ask "What time did you arrive in London?", a question to which "Last Thursday" is not a sensible response.
If I'm right about this, then "what time" is a two-word idiom that functions as one word, rather like "how much" or "how many" or the Spanish "por qué".
Huh. So, if it is an idiom, does this explain the absence of "at"?