Assuming the driver has the necessary permits to cross at Shenzhen Bay and enter into Hong Kong there shouldn't be a problem; fast modern highways connect the crossing with the urban area. Vast -- truly vast -- numbers of people cross between Hong Kong and the Mainland every day: it's no big deal.
I don't know anything about specific Hong Kong hotels or points plans, but in terms of area, for a one-night stay Causeway Bay would be a good choice: lively, especially at night, lots of shopping, and not quite a tourist ghetto like Tsim Sha Tsui. Also, set between Wanchai (another possible hotel location) and North Point, two other areas with a lot of character.
From Causeway Bay or Wanchai you can take a pretty quick taxi ride (or the MTR) to the Airport Express Station on Saturday morning, or just take a cab all the way to the airport, if your budget permits.