My mom and I just returned from Spain and a visit to the Alhambra. I was able to secure Alhambra ticket reservations from the US but only by using my American Express card. As others have stated Paypal is a good option as well.
Picking up the tickets - you definitely need your passport. That was the first thing they asked me for when I presented to the ticket counter up at the Alhambra. There are two other options to pick up your tickets down in town - those options will show up on your reservation print out. Neither of those options worked for me and since we had an 8:30am time slot I wanted my tickets in hand when the gates opened. So I ran up there the night before and picked them up. Just remember that ticket office hours changed on Oct. 16.
I also want to report out about the excellent first aid folks at the Alhambra who helped my mom during our visit. By the time we got to the Alhambra - it was our 10th day in Spain and my 83 y.o. mom was not doing well with her heart rate (we got caught in a Barcelona protest which didn't help). The morning of our Alhambra visit she looked particularly pale and she could barely get around the palace. When I saw the First Aid office near the Carlos V palace, I took her straight there. Sure enough her heart rate was very elevated. The EMT's at the first aid office called for an ambulance, got an IV started and took great care of my mom. They were especially kind given my very limited Spanish.
After a 10 hour visit to the hospital - mom checked out and ready to continue our trip. The cost for the treatment at the Alhambra, the ambulance ride and all the hospital tests and care - all free! Every single doc, EMT, nurse, tech and aide were kind and helpful. I work at a hospital here in the US and I was so impressed with the care and that every single person in Spain or visiting Spain is covered.