The train travel is difficult and long, so I don't recommend it.
There are small cruise ships from time to time, but they are neither cheap, nor fast, so you might inquire about them, but do not count on using one.
The buses from Varna to Istabnul are twice a day, according to one Bulgarian website, leaving at 10:30am and 9:30pm every day. The trip should take some 8-9 hours, I guess. The company operating the route is Metro Turizm.
is a very comprehensive travel website that lists other companies that supposedly operate the route, too. There are phone numbers in Turkey and Bulgaria listed. In any case, when you arrive in Varna, immediately look around for the buses; there will be at least two per day, I guess.
Hope this helps
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