Unfortunately, ferries from Sochi to Burgas and back yet. At the moment, the ferry service to Bulgaria is established only with Ukraine (port Ilyichesk), Georgia (Poti, Batumi). In the high season of this year, it is planned to launch a line to Turkey, the city-port of Zonguldak.

From Sochi, in turn, ferries go to Trabzon and Batumi. So, if you really want to arrange a sea voyage, at first from Burgas by bus or train you need to come to Varna. Next, you need to take a ferry to Batumi, and take a ticket to the ship in Sochi

Either you can get to the Turkish Zonguldak by ferry from Varna (when it will be launched), and already from it to move to Trabzon and then to Sochi.