I am not sure about the price but I would suggest you to take the subway.
With only one transfer from one subway to another, you can arrive at Sultanahmet faster than a taxi since you can come across a traffic jam depending on your flight arrival.
If you do have a lot of luggage, then ignore my suggestion.
None of the transport options (other than taxi) would be comfortable with a large luggage. about 7 years ago
If you would like a map of the metro/subway, here is one:
You should change/transfer in Zeytinburnu station. about 7 years ago
sas is correct. If you have more bags than you wish to haul on the subway, then a taxi is best. We were there last month and took the subway perfectly fine.
When you switch from the tram to the train at the station saw mentions, you are at the beginning of the line and will be able to ride the tram right to Sultanhamet.
We also recommend getting Istanbulkart. It will cost you 7 TL but you get 6 back when you return it. It will save you 12.5% each time you use it on fares and when you transfer the transfer cost is only 1 TL instead of the full 2 about 7 years ago