Ataturk International Airport is the main airport and close to Yesilkoy so is the same one you remember. It does have a very new terminal in the last few years.
It takes a good 30-40 mins from there to the cruise ship docks on the European shore if there is little traffic. For peak times like morn and eve allow more time.
As with many airports, beware of the taxis there. Unfortunately many may take you on for larger than the norm fares or bring you on a larger city tour before drop off. Try to get an idea of the fare or agree this before setting off.