A better way to phrase this would be, ‘Around what time should we meet?’ The person would respond with an estimated time interval to a relaxed question like this. Or, ‘What time should we meet?’ The person usually responds with an exact time to a question like this.

Yes, “About what time should we meet?” sounds very natural (at least here in the US).

I think for the question I would say 'What time should we meet?' The answer might be 'About ten o'clock'. Actually a more natural question is 'What time do you want to meet?'