We have a very early flight, so I'm thinking we should spend the night in the airport -- how difficult will that be?
You may not find anyone here who has personal experience sleeping in the Vienna airport.
But here's a website on sleeping in airports http://www.sleepinginairports.net/
There is a hotel Novotel right at the airport. You can get information about Novotel on Google.
I stayed at the NH Vienna Airport Hotel for the same reason. It is located directly across from the departure terminal, literally a few steps away. Used booking.com and had no problem.