I know you can purchase ticket from an automat in the Regionalbf, because I did it a year ago. I was going to the Mosel. I purchased a Rheinland-Pfalz-Ticket from one of the red/white/blue Bahn automats (w/ the touch screen) using a card, because that was all it took, then got my local ticket to Mainz from an RMV automat next to it and paid cash, because that was all it took.
There is also a ticket counter (DB Reisezentrum) in the Regionalbahnhof (see "Ebene 0").
If you go out to that "other station" (Fernbahnhof) you can buy a full fare ticket there for an ICE to Würzburg connecting to a regional train to Haßfurt. That will be the fastest way and cost €44,- per person.
The least expensive way, albeit slightly slower, would be to take an S-Bahn to the Frankfurt Hbf and a regional train from there. Most REs end in Würzburg and you have to take another regional train from Würzburg to Haßfurt. A few of the REs go on to Hof, stopping at Haßfurt on the way, so no need to change in Würzburg. You can ticket this as an RMV ticket (€7,15) from the airport to Kahl am Main, the first stop on the RE in Bavaria, and a Bayern-Ticket (Single €20, 2-5 people €28).