Istanbul Night Bosporus Cruise

Have you considered consulting Lale Aran at SRM Travel in Istanbul?

There are plenty of cheap local ferries that go on the Bosphorus, the Golden Horn, and on the Asia side - the locals use them, they are not for the benefit of mainly tourists. If at all possible (and it sounds like it may not be), go during the day, much better to see all the buildings that way. There is no need for a tourist-oriented tour, just look on the map and pick one that looks like it fits your available time. It's as simple as showing up and picking one. There are some tours that are specifically packaged for tourists and involve hanging out at a fortress up north on the Bosphorus for hours and having to eat at their predetermined restaurant. I picked a ferry that just went around and came back, it was totally worth it. Also, I loved going on there Golden Horn (different direction than Bosphorus) and to Kadıköy which is on the Asia side.

Maybe some of this info will help:
https://turkeytravelplanner.com/go/Istanbul/Transport/IstanbulFerry.html

By the way, you'll have to purchase a Turkish e-Visa just to leave the airport, but it's very inexpensive. I would get it online and avoid the lines at the airport.
https://www.evisa.gov.tr/en/

By the way, if you are flying Turkish Air, look into their Touristanbul sightseeing service. It's very time-specific and it may not work for you. It depends exactly when you get in and leave. They will also pay for your hotel in the case of a long layover. They do this to promote the country to visitors and to make Turkish Air an attractive option for fliers. It's free for the visitor.
https://www.turkishairlines.com/en-us/flights/fly-different/touristanbul/