Busses from Vienna airport to Südbahnhof train station operate between 6:00 am and 00:00 am. (There are also busses to other destinations in Vienna.)
Duration of trip: 20 Min from the airport to Südbahnhof,
cost: Euro 6.-.
This is the cheapest method, but you have to change trains once to get to Südbahnhof, see below. You can buy a ticket from Vienna airport (Wien Flughafen) straight to Graz, which costs only 50 cents more than the ticket from Wien Südbahnhof to Graz. Buy tickets at the ticket window or ticket machine on the platform.
Departures: Trains to Vienna from the airport train station Flughafen Wien operate between 4:54 and 22:39. Between 5:39 and 22:39 departures every 30 Minutes at xx:09 and xx:39. There are additional departures before 8:39 am To get to Südbahnhof, you have to change trains at Rennweg (22 minutes after departure from the airport).
Duration of trip: approx. 30 Min from airport to Wien Südbahnhof.
Cost: A single ride from the airport to Wien Südbahnhof is 3 Euro, but if you buy a ticket from Flughafen Wien directly to Graz, it costs only Euro 31.20 (compared to Euro 29.70 for a ticket Wien Südbahnhof to Graz).
Changing trains at Rennweg:
Get out at Wien Rennweg and for the last few hundred meters change to a
Schnellbahn train S1, S2 or S3 to Wien Südbahnhof and Meidling
(on the opposite track,
direction to Meidling - Liesing - Mödling - Wiener Neustadt).
There is one every 7 Minutes.
You can either get off this Schnellbahn at Wien Südbahnhof or continue to Wien Meidling. (All the trains from Wien Südbahnhof to Graz also stop at Meidling, at 4 Min after the full hour, i.e. 7min after their departure from Südbahnhof.)
The distance to walk from the light rail to the train line to Graz is shorter in Meidling, but you have to take the stairs down and up again through an underpass (trains to Graz leave from track 4 at Meidling); at Südbahnhof the distance is longer, but there are escalators. (At Südbahnhof, trains to Graz leave two levels above the main lobby (tracks 11-19), and the Schnellbahn light rail arrives one level below the main lobby.)
Taxis from Vienna airport to Wien Südbahnhof train station cost around Euro 25-30 (and there might be an extra charge for baggage). Duration of trip: 20 Min (longer during rush hour).
Duration of trip about 2h40min. Price of ticket Wien-Graz (one way) Euro 29.70.
You have a connection from Wien Südbahnhof to Graz every hour at 3 Minutes before the full hour from 5:57am through 8:57pm (5:57, 6:57, 7:57, ..., 19:57, 20:57).
Trains at 3 Minutes before the even hour (5:57, 7:57, 9:57, ..., 19:57) as well as the train at 20:57 are direct. (Destination of direct trains is shown as Graz, with the exception of 7:57 to Ljubljana and 15:57 to Zagreb).
Trains at 3 Minutes before the odd hour (6:57, 8:57, ..., 18:57) require changing trains at Bruck an der Mur (about 2hours into the trip): in Vienna, get on the train to Villach (or, at 6:57 to Silian, at 8:57 to Venezia) and change in Bruck/Mur to the train to Graz. (At Bruck, you will probably arrive at track 1 and then have to go through the underpass to track 3, where the train to Graz leaves 2 Min after the full hour.)
As the train climbs Semmering mountain, you will have a beautiful view on the left side (going from Vienna to Graz), between Payerbach and Semmering stations (between approx. 45min and 1h15min after departure from Vienna).
Cost (by public transportation - train or bus): Euro 1.60 (One zone in Graz's public transportation system. The ticket that you buy for the train or bus at the airport is valid for the whole ride from the airport to any destination in Graz, using any combination of streetcars, busses, etc.)
Bus lines 630 and 631 stop right outside the airport and go to Jakominiplatz (central bus and streetcar hub in Graz). There is a ticket vending machine at the bus stop where you can buy 10-rides (Euro 12.40) and 7-day tickets (Euro 7.90) in addition to the regular 1-zone ticket for Euro 1.60.
Duration of trip: 20 Min. to Jakominplatz.
Bus Departures: Busses 630 and 631 leave Graz airport for Jakominiplatz. A schedule is available here.
The train station is about 300m outside the airport. (A little shack visible from the airport's main exit.)
Duration of trip: 13 Min.
trains to Graz Hauptbahnhof leave Flughafen Graz station (daily, unless otherwise noted) at 5:12 am (not Sat/Sun) and between 5:54 and 21:10. An exact schedule can be found here. (With respect to days of operation, public holidays count as Sundays.)