The cheapest way is by Train(from Gara de Nord in Bucharest).
A single from Bucharest to Ruse(Bulgaria)costs 44lei(£9)-reserved .Takes about 2.5 hours.
To buy a reserved ticket,you have to go to the CFR office,which is located out side the train station,close to the Ibis hotel.Walk to the last desk on the right and the ticket can be printed for you there.
After arriving in Ruse get the bus to Varna.The bus station is right next to the train station,so you dont have far to walk.Go inside the main building and at the very end there is a girl there who will give you a bus ticket for Varna(cost is 16 lev=£6.50)and takes 3 hours.
On arrival at Varna bus station,just ump in a taxi and within 15 minutes,youll be in Albena.