You don't need to go somewhere else (by cab or similar) from the airport to reach a train. It has a station called "Flughafen Wien Bahnhof", which is easily accessible from the terminal buildings without even going outside any buildings. You exit to baggage claim, and then the way to the station is marked by signage. In the station, there is three platforms, the trains you will want to take run from 1 and 2. Getting to the station takes about the same time as you'd need to walk to a cab, so no extra time here.
The connection is an extra 30 minutes compared to what you already listed. This is the best option there is, possibly excluding taking a cab directly to Graz. There's no extra hassle and no extra risk, well, almost. You run a risk of full trains depending on the date and time, but that can be alleviated by making a reservation with your booking.
Details: Trains always leave at XX:33. For even hours you have to change at Wien Hauptbahnhof, for odd hours there is a direct train, both are 3h total.
You have 10 minutes to change trains at Wien Hauptbahnhof, the connection train leaves on the same platform directly opposite, no need to change levels, no problem catching the connection normally.
There are flex tickets, called Normalpreis, and fixed connection tickets, called Sparschiene. Both can be bought in advance, flex tickets can also be bought at the station. Booking in advance usually doesn't give you a better price for flex tickets.
You will want to book a flex ticket, those are valid for two days, on the day you buy it for and for the next day as well.
Concerning the policies for rail tickets if your flight is delayed, you are out of luck if you booked a fixed time ticket, and you are late for the train. You would have to buy a new ticket, and lose all the money spent for the first one.
Depending on the arrival of your flight and how much time you have, you could also take a flex ticket to Vienna and then a fixed ticket in the evening or on the next day to Graz. This would reduce the total ticket price, and some sightseeing in Vienna is definitely something I'd recommend!
You can use tickets.oebb.at to book in advance for all kinds of tickets. They also have an app available from Play Store and App Store. Booking via the website allows you to reserve a specific seat, which is not possible in the Apps. You can print your online ticket or show it in the App, regardless of how you booked.
There is also the option of getting a ÖBB Vorteilscard for half price train tickets, it's valid for one year and costs € 66,- if you book it online. Depending on how much you travel, it might save you some money, or not.