I suppose that might have been possible in the UK, but back in 1929 in the US, stamps weren't available nationwide on the first day of issue. You had to send requests to the first day of issue post office and they affixed the stamps to self addressed covers for postmarking. Therefore, a 1:00 AM postmark is quite unusual (I think).
Anyway, if you want to consider the other extreme, here's another US First Day Cover, this time with a late date stamp -- 11:30 PM and on New Year's Day (January 1, 1932). I think this would have been about the latest date stamp possible for a First Day Cover!
