You are correct, US citizens no longer need a visa. I agree with the recommendation to get private transport from the airport to your hotel. I used Welcome Pickups shuttle and it worked great (I only did pick up but you can save $ booking a return). It is 45-60 minutes from the airport to the city center, traffic is simply a nightmare in Istanbul, no way around it (and it's longer on public transit).
What is your actual flight arrival time? Even if it is on time, the airport is freaking enormous and some flights taxi for at least 20 minutes to the terminal (mine took a little longer). If you're checking luggage that will take some time as well. So if your flight arrives at 4 p.m., taxiing, border control, luggage, and shuttle will get you to your hotel maybe 6 p.m. at the very earliest, probably more like 7.
What day of the week are you arriving? Many Turkish baths are open later hours but you'd want to make a reservation to be sure. With such limited time, I probably wouldn't select that as an option. Both the spice market and Grand Bazaar close at 7 p.m. most days (grand is closed on Sunday). Hagia Sophia has a new process with entry fee and closes at 7:30 p.m. It's about a 15-20 minute walk from your hotel to the Hagia Sophia area. I'm not familiar with the area of your hotel, but you could walk there, look at the outside of things, have dinner, and return to your hotel to get some sleep (you should want to do that.)
Welcome Pickups (and probably others) has a sightseeing tour of sorts which might be an option for you in the morning you return to the airport - they essentially drive you by things but you won't have time to go in. Given your very limited time you might also consider hiring a private guide who can efficiently get you to places you want to see. As acraven suggested make sure you know how much advance check in time you'll need for your onward flight (traffic, security, etc.)
Best of luck!
