If you're looking for the cheap way to get from the airport to the Hotel Bristol, find the train station in VIE (it's pretty well marked), take S7 into the city (I believe it runs every 30 minutes, perhaps better now), get off at "Wien Mitte - Landstrasse", transfer to U4 heading to Huetteldorf (U4 is the green line at Landstrasse), go 2 stops to Karlsplatz/Oper stop. The Bristol is right across from the Opera House, and it's about a 3 minute walk from the U4 stop to the Hotel. Fare should be around $4, instead of much more for the taxi.
Instead of S7, there's also a bus that takes you to Wien Mitte (City Air Terminal) which is right outside of the Landstrasse U4 stop. The bus comes every 20 minutes and it's a little faster than the train. Personally, I prefer the train.
Hope that helps!