There are only three companies operating scheduled flights directly from Edinburgh to Alicante:
Ryanair will fly you out on 21/09, and back a week later, for a total of �127.87 per person including taxes and charges (but excluding baggage check-in fees and card handling fees).
http://www.ryanair.com/site/EN/
easyJet don't have flights on Sundays on that route. Flying out on Saturday 20/09, and back a week later, is currently priced at �137.77 per person including taxes and charges (but excluding baggage check-in fees).
http://www.easyjet.com/en/book/index.asp
FlyGlobeSpan are dearer, with a total price of �171.98 per person.
http://www.flyglobespan.com/
Airtours, Direct Holidays, Escapades, Holidays for Less, JMC Holidays, Sunset Holidays and Thomas Cook all book passengers onto the charter service which operates between Edinburgh and Alicante, but this is only available on Tuesdays. (There are no other charter flights available).
So it looks like Ryanair might be the cheapest option.
Chris