bayern ticket

You need a Airport City Day group ticket (23,20€) Valid for travel on the S-bahn from the airport and on all munich buses, trams, S-Bahn and U-Bahn for the day. If you are planning on travelling on to Salzburg on the day you arrive at the Airport then the Bayern-Ticket (38€) is the right option for your group.
http://www.mvv-muenchen.de/en/tickets-fares/tickets/day-tickets/airport-city-day-ticket/index.html
http://www.munich-airport.de/en/consumer/anab/bahn/index.jsp
https://www.bahn.de/p_en/view/offers/national/regional/laender-tickets/bavaria-ticket.shtml

If we are traveling to Salzburg the same day as arrival do we still need the Airport City pass, or does the Bayern ticket cover everything?

You only need one or the other, so choose the Bayern ticket if all the travel is on one day.

I assume you are arriving at the airport in Munich and going directly to Salzburg the same day rather than spending a night in Munich.

The Bayern-Ticket is valid for the entire day (from 9 am workdays until 3 am the following day) and covers all transport of the Munich metro (MVV) and regional trains all over Bavaria as well as outside Bavaria to Salzburg, which is a border station for the Bayern-Ticket, so, yes, your entire trip from the Munich airport to Salzburg will be covered.

However, don't take the S-Bahn to the Munich Hbf. Take the S8 S-Bahn from the airport to Munich east station (Ostbahnhof) and change there to the regional train to Salzburg. It's faster that way. At the airport, the S1 gives it's destination as Ostbahnhof, but that is the final destination, after it has gone into Munich from the west end and gone entirely through the town. The S8 goes into town from the east, through Ostbahnhof, then through town to a stop on the west (usually Herrsching or Weßing). The S8 will get you to Ostbahnhof sooner.

Once in Ostbahnhof, go through the tunnel out to track (Gleis) 8, where the train to Salzburg stops.

Thanks for all the great recommendations, you have made our trip less confusing.