hi :)
as far as I know there’s no way directly to reach Istanbul by sea from Burgas, there are no such direct ships or at least I’ve never heard about such kind of thing. It’s very close to Bulgaria and the most common way is by bus- buses depart to Istanbul from Burgas every day and cost about..let’s say max 25-30Euro. The buses depart from Str.”Bogoridi” in Burgas- at 12.00 and 19.00 BG time. I’m sure that there are other hours from other companies but I don’t know them exactly. When you are in Burgas you can ask for buses departing to Istanbul in “Bulgaria”hotel, it’s in the center of the city, no way to miss it.
The travelling time is about 8-10 hours, it depends how long will you stay at the border.

Let’s hope that someone else knows something about ships to Istanbul :)

Here I can give you a link with the buses Sofia- Istanbul, just in case.

http://213.240.239.190/guide/searchResultDepartures.jsp;jsessionid=0000Og36GIq75TVfVoaBXjzPX6K:-1?simple=1&direct=0&quickPointId=619&t1=&endPointId=664&period=0&search=

All the best