Public transport in Vienna is convenient and safe. Trams and underground trains generally run until midnight on weekdays and throughout the night on weekends. Information on first and last trains is signposted at each station. There are also night buses.
Inside each bus, underground train or tram there is a list of all the stops on that line. The routes with all stops are signposted at bus and tram stops.
Route & travel planner: http://efa.vor.at/wvb/XSLT_TRIP_REQUEST2?language=en
Ticket vending machines take Euro-notes and credit cards but it is advisable to have some small change on you. Zone 1 ("Kernzone") covers all of Vienna, while Zone 2 also covers the outskirts of Vienna including Vienna Airport (Schwechat). Vending machines are available in all underground and train stations and can be switched to English and several other languages. A single trip inside Vienna costs 2,10€ and can be bought by pressing the "Einzelfahrt Vollpreis"-Button. Tickets are also available at newsagents, tobacconists ("Trafik"), or ticket offices at railway stations. Single trip tickets are also available but more expensive on trams and buses (note, you will need coins for the vending machines).
Ticket information and online purchasing: http://shop.wienerlinien.at
Taxis from the airport (http://www.viennaairport.com) to the city centre are expensive. Limousine service can be ordered online, for example from http://www.airportdriver.at (33€).
If you prefer public transportation you can either take the CAT train (online price 12€ one way, 19€ return) to "Wien Mitte, Landstraße" or use the regular train S7 in the direction "Floridsdorf" to "Wien Mitte, Landstraße". For this trip, you need a 2-zone ticket (4,20€ -- for example, type in Wien Spittelau on the vending machine display) which covers transport to your destination within Vienna. The S7 train leaves from the airport every 30 minutes. Tickets are available from a vending machine on the platform from which the train leaves.
"Wien Mitte, Landstraße" is a central train and subway station from which you can easily reach all places in the city center such as the institute or your hotel. For Hotel Anna Katharina it is best to take the U4 to Spittelau, take the south exit (Nordbergstraße), and walk the remaining 200 meter.
Location of the Mathematics Department on Google maps
The best means of public transport to get close to the Mathematics Department are subway line U4 (stop Roßauerlände or Schottenring) and the D-tram (Stop Schlickgasse). From U4 stop Roßauerlände resp. Schottenring walk 200 meter south resp. north to Oskar Morgensternplatz. The mathematics department is in the 12th storey high part of the modern building at this square (opposite to Roßauer Bridge) or Schottenring). The D-tram stops at Schlickgasse, opposite a Billa supermarket. Walk in that direction (along Berggasse) for 200 meter to reach Oskar Morgensternplatz.