The metro is arduous! I have done this recently with a friend who believes he will have his organs harvested if he takes a taxi or any private transport overseas!
We were staying on Istikal Street, nearest station Sishane.
But back to the metro. It was a long trudge to the airport metro station from the terminal Take the lifts down otherwise its escalator after escalator after escalator. There are long waits between trains...I waited 25 mins. I did 6,000 steps from getting off the plane at IST to boarding the airport train - they literally couldn't have made me walk any further!
We took the metro to the end of the line at Kagithane (line M11 airport metro).
Then you must change to line M7. This involves going up to the street, through the barriers and you have to cross the street (maybe 200 metres)...2 pedestrian crossings as the stations are not in any way connected (yet). You go through the barriers, paying again. If it's raining...you will get wet!
Then you take the train to Mediciyekoy and change again to the M2 line. Again, you go all the way up, and through payment barriers again then all the way down again...it is a long trek but at least not at street level this time!
It looks like to Sultanahmet you'd do all this...AND then another change....so 3 changes all up.
For the airport to Sishane it was minimum 90 mins when connections were quite well aligned. You essentially pay for each of the trips separately, so the fare is not as suggested by Google maps...it's about double that, but still less than A$3!