EDIT
Turns out using Stagecoach for the whole trip from EDI to my hotel in Anstruther was easy, comfortable and inexpensive. No need to reserve--get on the bus and pay. A change at St. Andrews and the bus stop in Anstruther was a block away from my hotel. They seem to have a variety of coaches, but all have a luggage hold underneath.
After I land in EDI next week, I plan to take the 787 to the bus station in St. Andrews, and take a taxi or Uber from there to my lodging in Anstruther.
Is there a reason to buy the ticket now, or is buying it on the bus ok?
Are they often full?
Is Uber a possibility in St. andrews?