Munich Airport to Salzburg

The airport is connected to the city by Munich suburban railway (S-Bahn) lines S1 and S8. The ride takes approximately 45 minutes to the Marienplatz station in the city centre.--where you'd swap to the train to Salzburg. The S-Bahn from the airport to the city runs approx. 20 hours a day with a short break between 01:30 and 04:00 a.m.

By early morning, what time is your scheduled arrival and how many in your party? A Bayern ticket may be the way to go. Trains to Munich alternate between the S1 and S8. If you get the S1, change at Hauptbahnhof, if you get the S8, change at Munich Ost.
Don't change at Marienplatz, the are no trains to Salzburg to be had there.

Thank you for the response. Would it be easier to take a bus from the airport to Salzburg? Is that possible? We arrive by 8:00 a.m. to Munich.

The Meridian regional trains from Munich to Salzburg run hourly. They start at the Hauptbahnhof and stop at München Ostbahnhof (Munich east station) a short time later, then go on to Salzburg. Each S-Bahns run at approximately 20 minute intervals; the S1 goes first to the Hbf, the S8 goes first to Ost. You can catch the hourly train to Salzburg using either S-Bahn, but the last chance to catch a given regional trains is using the S8 to Ost. That's why I recommend that connection.

The earliest you can use a Bayern-Ticket for travel is 9 AM on workdays (like Sept 1). There will be an S8 leaving the Munich airport station at 9:04. It will get you to München Ost at 9:36, with plenty of time to catch the Meridian train at 10:04. The next S-Bahn to leave the airport is the S1 at 9:11. Don't be tempted to take that train. It arrives at the Hbf at 9:57, 2 minutes after the Meridian train leave the Hbf, and it arrives at Ost at 10:06, 2 minutes after the Meridian train leaves there. Wait for the 9:24 S8. It gets to Ost at 9:55, giving you 9 minutes to catch the Meridian train.

BTW, don't be fooled by the departure board. It will say that the S1 is going to Ostbahnhof. And it is; that is it's final destination. However, it goes into Munich from the west, stops at the Hbf, then goes through town to Ost. The S8 will show a final destination of Herrsching or Weßling or something else, but it goes into town from the east, stopping early at Ostbahnhof, then goes through town to it's final destination. The S8 takes less time to get to Ost.

The train that leaves Ost at 10:04 gets to Salzburg at 11:41. If you can get to the airport S-Bahn station by 8:24 (or 8:04 or 7:44), you can get to Ost in time to catch the 9:04 train, which gets to Salzburg at 10:41. You can't use the Bayern-Ticket before 9, for the S-Bahn, but it will be valid for the 9:04 train. If the hour is important to you, you can pay separately for the S-Bahn (in addition to the Bayern-Ticket) and start your travel earlier than 9 AM. An MVV single trip ticket from the airport to town (4 zones) is 10,80€ per adult. Buy the single trip tickets, as well as the Bayern-Ticket, at the airport . If you can't catch the 8:24, wait until 9:04 and use just the Bayern-Ticket.

If you arrive at 8 am, by the time you exit the plane, walk to the immigration station and wait in line to get your passport stamped, then wait for your luggage to come off the carousel then pass through the green door into the airport, it is liable to be very close to 9 am. So I would get the Bayern ticket. 23 EUR for the 1st person and 5 EUR for each additional person up to 5 traveling together. I just heard you say "we", so there must be at least 2 of you. If so, then it would cost 14 EUR per person, if you buy it out of a train ticket vending machine. You can buy from a person at a ticket window, but it will cost just 2 EUR more for the transaction fee. So as Lee pointed out, you can take the 9:04 S8 to Munich Ost and change there to the 10:04 Meridian train to Salzburg. If you are running a bit late, you can still make the Salzburg train by taking the 9:24 S8. These will have you in the center of Salzburg by 11:41.

I don't know of any buses from the airport to Salzburg. The train service is very convenient and easy to use and will get you to Salzburg every hour.

Flexibus has trips from Munich airport to Salzburg Hbf, but it takes longer (3 hours), costs more for two adults, and only runs a few time per day.