Logistics of a 2 AM arrival in Istanbul

Does your hotel have a 24 hr front desk? If so, then I cant see why you couldn't check in in the middle of the night. Add an extra night onto the front end and let the hotel know of your very early/late arrival time.

Galata's Hotel has a 24 hour front desk and it is not unusual for guests to arrive at that hour. To get in your room immediately, you'd need to book the hotel for the day before you arrive and let them know that you won't check in until 3 a.m. at the very earliest (probably more like 3:45 it takes time to exit the airport and 45-ish minutes to get to town even at that hour). The hotel offers door to door private transfers, but you can also book your own using something like Welcome Pickups. You could also take a taxi, traffic won't be as bad in the early hours as it is mid day.

You may have better luck calling the hotel. I typically find day time front desk staff sometimes speak more English, so check the time difference from your home time zone and try calling a bit after their check out time (but before check in time) when the staff isn't preoccupied. If you're staying at the hotel with a RS tour make sure they know that so you aren't faced with changing rooms. And say hi to Princess the hotel cat, she's a spunky one!

As above, just phone your hotel and book an extra night, notifying themnkf expected arrival time. I do this fairly often, and it's the simplest solution.

At the airport-

iGA Napzones: The airport provides free, semi-reclined lounge chairs in designated "Napzone" areas across different gates (A, B, C, E, F, and G). These are first-come, first-served and are located in the quieter sections of the terminal.