score:3
For what it's worth (I know you're going the other way) but the Edinburgh Airport website says you can buy tickets from the information desk -- I've no idea if they take cards.
As Tor-Einar Jarnbjo says, Travelshops are your best bet -- also note that the Waverly travelshop is closed until 'Spring 2014', they will accept cards but I don't know if they'll charge a fee for doing a transaction that small. However, I don't know where you're staying, but hotels and hostels will print stuff for you (they may charge) and there are plenty print shops lying around that will do it.
If it's about saving money then note that the Airlink is relatively expensive at 3.50GBP (but fast), you can save money by taking the normal number 35 bus (1.50GBP single, any distance). It'll depend where you start from, but if you're near Waverly just wander along to the bridges to get it -- if you're nearer Haymarket it might be a bit too much of a trek. The 35 takes about an hour to do the full route, but if there's traffic or roadworks that can be longer -- and it makes a lot of detours and stops along the way. I have very nearly missed a flight relying on that bus, so leave more time than you think you need.
Upvote:4
I am not 100% sure, but I think you can pay with card if you buy your ticket in one of the Lothian Buses' Travelshops. There is at least a Travelshop at Waverley station if you plan to take the bus from there.