Remember that Uber offers (at least most places) a way to reserve in advance for a ride. With your particular requirements, it might be less stressful to book your needs in advance. It's much more expensive that way, but at least you don't need to stress over availability getting your ride with the exact type of vehicle you'll need and the time you need it.
I also agree with other posters that drivers should be available at that time. Early morning is probably a popular time to guests needing a ride, so drivers will be available.