1

The S-Bahn is EUR 4. A taxi would cost you around EUR 30, and will give you some half hour extra.
For an in between flights visit I should say just do some walking to get the feel of the city. Real visiting is no option, when you realise your effective time in the city will not be much more than eight hours.
The Ring with those mighty late nineteenth-century public buildings is one thing you should not miss. And just wander around in the city centre, have a look at the Postbank and the Secession building to see how revolutionary Viennese art was at the turn of the century, go for a coffee and a torte in the Cafe Central, and then head to the Prater to see the Riesenrad, famous from the 'The Third Man', and then slowly look for a restaurant before you go back to the nearest S-Bahn station for the airport.

2

My advise would be to get the S-Bahn (around €4) from the airport to Wienmitte station - do NOT get the City Airport Train, it's a total rip-off. The S-Bahn goes to the same station and takes only 9 minutes longer! From Wienmitte, you can either walk through the Stadtpark to Stephansplatz (the centre of the city) or hop on the U-Bahn and maybe go to Karlsplatz or Museumsquartier (Museums, beautiful buildings etc) - the Karlskirche and Wien museum on Karlsplatz are definitely worth a visit. Lunch in Naschmarkt (nearest U-Bahn = Kettenbruckengasse or Karlsplatz) is great - good atmosphere and lots of choice.

Also - things at Vienna airport are really efficient. It's pretty small - luggage comes through quickly, passport control is quick (or non-existent if you're coming from the EU) and the whole thing takes literally 10 minutes from getting off the plane to exiting the terminal.

Have fun!! If you want any more tips for things to see, let me know :)

3

use the "CAT" (16 Euro) its the fastest way to vienna, arrive in "WIEN MITTE" , get a map of the city and start walking towards "STEPHANSDOM" then walk through the first district, then a late lunch for a rest , continue to "MICHAELERPLATZ" ,"HOFBURG" you can go by tram around the "RING" its the street that circles the first district. Have coffee and a delicious cake in one of the "AIDA" outlets, its a coffeehouse chain in Vienna,
the post number 2 suggested "MUSEUMSQUARTIER", a great place to visit as well, maybe "SPITTELBERG" christmasmarket behind "MUSEUMSQUARTIER"
after all that you will crash down on your way back to the airport.
there is also an airportbus to "SCHWEDENPLATZ" something like 7Euro

4

Is the Cat the hop on hop off bus? If so it's a good way to do a quick circuit without getting off then returning to any point on foot that looked interesting to you. nowornever's tips will also do nicely!