Hi (first post for me)
The cheapest, scenic and most enjoyable transfer from Sabiha Gokcen (SAW) to Sultanahmet is by by public transport. Takes between 1.5 and 2 hours.
Directions are:
Take the E10 bus (green) from right outside of Arrivals. Leaves every 30 minutes, and costs TL3.5 (pay the driver).
The trip takes between 40 minutes and an hour, and stops to pick-up and drop-off several times. The driver doesn't hang about, so make sure you get a seat.
Get off at the terminus at Kadikoy. You can't miss it, as it's also the ferry terminal (make a mental note of where the bus stop is for your return trip).
In front of you will be the Sea of Marmaris. To the left (100 yards) you'll see the ferry terminal. Go in to the terminal and buy a jeton from one of the machines just inside, costs TL1.75. Then go through the barrier. You want the ferry to Eminonu (every 20 minutes or so). An illuminated board shows which ferry is which. There's only one stop, ie Eminonu, and it takes about 25 minutes (smashing little cruise, and a great introduction to the awesome city that awaits you!)
When you get off you need the tram, the stop is just to the right as you look at the road. Go to the tram stop and you'll need to buy another ticket (jeton), again TL1.75).
The tram will get you up the hill to Sultanahmet in about 5 minutes. The stops are named, and Sultanahmet 3rd stop) is the next stop after Gulhane.
If you are travelling relatively lightly it's a very pleasant 15 minutes walk from the ferry, just follow the tram line.
Sounds complicated, but it's really easy, and the total cost is TL7!
The last ferry from Kadikoy to Eminonu is about 2030, after that you can get the ferry from Kadikoy to Karakoy (right opposite Eminonu, just over the Galata Bridge), and get the tram from there.
Details for the airport shuttle from Sabiha Gokcen are
www.istanbulairportshuttle.com, or phone: 00 90 212 518 03 54. Takes an hour or so depending on traffic. Cost 10 euros each way.
Some good and inexpensive hotels in Sultanahmet are Hanedan, Peninsula and Grand Peninsula. A couple in Gulhane (stop before Sutanahment) are Sude Konak, and the Agan. But there are 100s to choose from depending on your budget.