Hi, in my opinion, the underground is one of the most comfortable means of transport in Vienna... The lines U3 and U6 reach Westbahnhof... While the nearest underground station to Staatsoper is Karsplatz which is reached by the lines U1 and U4, so I suggest you take the U3 and you reach Stephansplatz, there change line and take the U1 ( the next stop is Karlsplatz)... Otherwise you can also reach the Staatsoper by walk from Stephansplatz, it is not very far...
If you need a map of the underground:
http://homepage.univie.ac.at/horst.prill…
http://aesop2005.tuwien.ac.at/pix/accom_…
The only problem is that it is not very panoramic... But I don't know if there are any busses that could be useful to you... I know that there is an S-bahn ( a tram), but it doesn't reach the Staatsoper...
Good Luck ^^
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If I'm not wrong, a single ride ticket (Einzelfahrschein) costs 1.70 euros
If you prefer a daily ticket (24 Stunden Wien - Karte), it costs 5.70 euros
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Regarding the time, I think you'll need 10-12 minutes about from Westbahnhof to Stephensplatz and 5 minutes from Stephensplatz to Karlsplatz (by underground)
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You're welcome ^^