Information about public transport in Vienna:

•    You can go by train or bus from the airport to the city. You can use the CAT (https://www.cityairporttrain.com/de/home) or the conventional train (all 30 minutes - much cheaper ~ 5€) or buses to different sites in Vienna (~8€).
•    Vienna has a very developed public transport system and many specific offers (see https://www.wienerlinien.at/eportal3/). You also will find a route planner at this homepage).

If you buy the ticket for public transport in advance (online), you only have to take the railway ticket from the airport to the city border (which is € 1,70).