I went on this tour last September. Your best option is a private transfer (Welcome Pickups) or another option is a taxi but only with a pre agreed upon price. I used both during my travels last fall. Additionally, RS tour company always provides recommendations.
For whatever method you chose make sure to identify the numbered meeting point, gate or exit door. The entrances/exits to the airport are guarded. Once you exit through the terminal doors for transportation, shuttles, taxis, transfers, etc. you cannot re-enter at that level. You’ll have to walk the long sidewalk to the intended meeting point. Or you must make your way upstairs to an entrance door where security screening is available then back down again. The terminal entry screening is low key.
Getting to Sultanahmet from Galata will be a breeze.