Croatia is NOT a member of the Schengen zone yet, a Schengen visa is therefore "useless". Check the visa requirements for your citizenship!

PS There are plenty of both cheap and expensive non-stiop flights from FRA to IST. Why do you want to choose having an overnight connection? By the way, Dubrovnik, a relatively small city, is so overrun by tourists in the summer that, if you find a hotel for one night, this may be several kilometres away from it!