CAT is currently not running. Is there any local information as to when it may resume? I've always just used CAT in the past. Also, we will have an early departure (6am), would booking a taxi be far more practical?
Yes I saw that recently. But I was also researching other options and apparently now there are both S Bahn and OBB trains that run to the airport, each twice an hour. Or a bus. Otherwise pre-arranged taxi is always an option, especially at the obscene hour you will need to be at the airport.
The CAT is a rip-off anyway and should not be used. It is targeting gullible tourists. There is only one reason to use the CAT, i.e. if you use the city check-in for your flight and drop your bags at the CAT station.
There are two train routes departing from Vienna International Airport. Which one to use depends on your final destination in Vienna.
The S7 trains (Schnellbahn) go to "Wien Mitte" train station, connections to Metro (U-Bahn) lines U3 (orange line) and U4 (green line).
The RJ and RJX trains (RailJet and RailJet Express) go to "Wien Hbf" (main train station), connection to line U1 (red line), and to "Wien Meidling", connection to U6 (brown line).
wmt1 is 100% correct. The demise of the CAT is a good thing as it was really a rip off (except for the noted exception noted above). It wouldn't have worked anyway with your 6 am departure. You should just take a taxi (about 36 Euro) or Uber (similar cost) - much easier anyway and takes about 15-20 minutes at that hour.
Due to the Corona crisis passenger numbers of Vienna International Airport dropped by 90%. This was also a major blow for all taxi companies. They try to get back into business offering very competitive prices (starting from €25).
This is a comparison chart, but there are about twice as much companies as listed here:
We used this company in the past, but I don't know how they managed during the past year: