Bus from Santorini Airport

Thanks for asking this question. We are going to Greece in the fall, so it's helpful for me, too.

• I found this link -- http://www.ktel-santorini.gr/ -- to the bus schedule. This is the one that is online today. It's a little odd-looking, but there are clickable links at the top, including one to the schedule.
• At the airport website, I learned that for public transportation, the bus is the only way to go (http://www.santoriniairport.com/to-and-from-airport/santorini-airport-public-bus/). It says the following:
  

"There is only one mode of public transportation in Santorini, the local bus system. It should be noted that there are no buses traveling across the island, going from one side of the island to the other.
Every bus goes to the main bus station at Fira (the capital of the island) and any changeovers for the next bus destination must be made from the main station only. The journey from the airport to Fira takes about 20 mins and costs EUR 2.20 each way. The bus service does not run on a regular basis and may only run every few hours (i.e. 7:30am, 11am, etc), especially in winter. The bus schedule is posted outside the airport and just before the exit. TIP: It is very handy to have a copy of the bus schedule or write down all the departure times and keep it with you, so you know when the bus will come."

• I found no mention of any way to schedule ahead, and I wouldn't expect one since the cost is so low.
  
• Perhaps they were talking about the Shuttle and Private Transfers services. You can click that link at the airport website to the left of the public transportation link. And of course, there are taxi and car rental options.

In 2008 we flew to Santorini and stayed in Oia. We contemplated using the bus but since this was the end of a 20+ hour travel day opted to use a taxi. As I recall, the charge was about 20 Euro. It was well worth it because I am not sure how we would have found our place in Oia after dark since the address is simply Oia! During our stay on Santorini we did use the bus system. It does operate as a hub out of Thira. There was a small building with schedules posted for the various busses to island locations. Taxis are plentiful on the island. I don't recall any other form of public transport so don't know how you would book anything in advance.