If you can get Munich to Graz included on you flight ticket, that would be the best solution. That way, if your arrival in Munich is delayed, they will just put you on the next flight to Graz, and there are several more that day. But that's too easy; you probably already thought of that. I suspect you are using United miles, and they don't include Graz. If you book Munich to Graz as a separate ticket, you'll have to retrieve your luggage and go through customs in Munich, then recheck your luggage. The airlines might not honor the Munich to Graz ticket if you miss your flight due to a late arrival in Munich.
Barring flying to Graz, I think Russ has a good alternative. Using a Bayern-Ticket to Salzburg gives you complete flexibility the first day, and spending the night in Salzburg breaks up an otherwise long train trip. You'll arrive in Graz at noon instead of 6 or 8 PM the previous day and really only miss the morning in Graz. Whenever you arrive at MUC, go to the S-Bahn station and buy a Bayern-Ticket. If you can't understand the ticket automat, you can buy it for 2€ more at the Bahn counter in the Munich Airport Center (MAC) upstairs or online. Take the S8 to the Munich East station (München Ostbahnhof). Note: the S1 will say it's destination is "München Ost", but that is it's final destination. It comes into town from the west and goes entirely through town to Ost. The S8 will say something like "Herrsching", but it comes into town from the east, goes to Ost, then goes entirely through the town and on to it's final destination. The S8 gets to Ost faster than the S1. At Ost, go out to track 8 and catch the next hourly regional (usually Meridian) train to Salzburg.
Another option, which I, myself, would use, if you can manipulate the Bahn website, would be to input from: Munich Airport T, to: Graz, with a stopover in Freilassing for 18:00. Since you are traveling the second day to Graz from Freilassing, which is across the river from Salzburg but still in Germany, the Bahn will offer you a 58€ (for 2) SparPreis fare for the Munich airport to Graz using EC111 from Freilassing at 9:51 to Bischofshofen, then IC 515 to Graz at 14:14. Because of the Bahn's Vor- und Nach-Lauf rule, you are not restricted to the specific regional trains on the ticket, but can use any regional train on that route all day on the 30th. The only specific trains are the EC to Bischofshofen and the IC to Graz.
Besides saving a few euro, this option would solve the problem of finding economical lodging within walking distance of the Hbf in Salzburg. I've stayed in Freilassing three times. The first time was here. Note: book directly. Booking.com is more expensive. Hotel Krone is a 6 minute walk (about 600 yds) from the train station via the tunnel under the tracks.
Another place in Freilassing is here. It's a little farther away (10 min walk) from the Bahnhof, but less expensive. It's essentially a restaurant with a few rooms to rent in back and upstairs. I've stayed there twice. It's nicer inside than it looks from the outside. The website is entirely in German, but Herr Chudoba speaks good English. Email him at [email protected]. If you want to go into Salzburg for the evening, a bus to the Altstadt stops on Muenchner Str, a block away from Hotel Rupertus. You won't find Hotel Rupertus on any booking website.
