There are regular flights of the Bulgaria Air every day from Sofia to Varna as well as many charter flights from Europe directly to Varna. To get from Varna Airport (via town of Varna) to Albena (about 40 km) one can use the regular public transportation between 7 am and 7 pm, and by taxi only outside of this time interval.
For those who intend to fly to Sofia there are buses and trains every day from Sofia to Varna as well as convenient direct bus lines from Sofia to Albena (about 450 km). The travel is about 7 hours. The distance from Sofia Airport to the Central Bus Station of Sofia is about 15 km and the best option is to take a taxi (the fare is about 10 EUR).
There is a regular bus connection from the Central Bus Station in Istanbul (Turkey) to Varna and vice versa. The travel is about 8 hours.