We have 13 hours layover at Istanbul while traveling to Europe by Turkish Airlines. We don’t have a transit visa. Can we book a room night at any hotel within the airport premises?
There is the YOTEL Airport Hotel that has 174 rooms available only to airside passengers. Rooms are bookable online and can be booked for as little as 4 hours.
As the airside part of the hotel is only accessible to transit passengers you should be fine to use it without a visa.
I assume you're Indian ? If you have a schengen visa, you can get a visa on arrival at Istanbul (you have to apply online - its a simple process) and get the visa stamped against your online visa receipt. It will give you time to explore a bit of old Istanbul (the main sights are close together) and have a better choice of hotels.