I will soon arrive to Istanbul Ataturk after midnight. The plane will land 00:45, passport control and bag collection will take at least 40 minutes, which means that I have to catch a bus or metro after 01:30. So, what are the available options to transfer to Sultanahmet, since my hotel is there. Note; I rather not take the taxi, so it will be my last choice.
The Havatas bus runs 24 hours. But the bus only goes to Taksim which from your timing you won't be able to get the metro to Sultanahmet from Taksim Square.
If the hotel that you will be in has a taxi service I'd suggest you to call them up and order a taxi. Or else, getting a taxi from the airport would be around 100TL. I took taxi all the time when I was working in Istanbul and I got no problem at all with the taxi. So I think with your timing I think getting a taxi from the airport would be the best in your case.
However, at Taksim there will be taxis waiting for you all the time especially Friday and Saturday night.
There is a transfer compny called Efendi Travel whose office is located in the Old City. They also work 24/7 at the airport for transfers.
You can make an online reservation via their website and they make your transfer from the airport to the exact point you want to go.
The updated prices for Sultanahmet (from the Ataturk International Airport) is approximately 20-30 USD and you will have a private car for up to 4 passengers.
I never advise tourist to take taxi because some of them make your trip to your hotel longer in order to receive more money than it costs. So for tourist taxi must be the last option in Istanbul.