I was wondering if anyone had done this via public transportation...some combo of bus/boat/train? Could you say the route you took and if it was it a pain to do? (that is would it be better just to pay for a private transfer?)
Based on wikitravel, there is a Havas bus that takes an hour that goes to Taksim from where you can easily grab a bus or cab to Sultanhamet.
http://wikitravel.org/en/Istanbul
The bus schedule can be found here
http://www.havas.net/en/shuttle-parking/istanbul-sabiha-gokcen-airport/
hi
,this is Gokhan from istanbul Turkey,
there is shutle in every 30 minutes between SAW to Taksim centre ~10 Euro and takes 45min up to 1.5 hours upon tio the traffic
Than you can take a taxi for Sultanahmet ~10€ (max. 20 TL) in 10 or 20 minutes
Join to be in Turkey,,,
I took the city bus to the nearest metro station (don't remember what the stop was called, sorry), then took the metro to Taksim square and got the bus from there. It did take quite awhile, but was not that hard to do. But if you have a lot of luggage, are in a hurry, and/or are particularly tired or overwhelmed, I would recommend taking the airport shuttle bus, but again those usually go to Taksim square.
Or if you're and "intrepid traveler," traveling very light and arriving during the day, you can actually walk from Taksim Square to Sultanahmet, which is what I did every time afterwards. It's a long walk but takes you through/past the majority of the stuff you've probably come to see. You just walk down Istiklal Caddesi forever and ever, cross the bridge and veer left on the other side, toward Hagia Sophia/Blue Mosque.