Just returned from (un)exciting Almaty two days ago. Yes there is an airport hotel a stone's throw from the terminal. I stayed in a no-frill single and was happy with the convenience and the value. There is a 24 hour cafeteria downstairs which is reasonably priced. During the day, you can take bus 79 from near the airport entrance (5 min walk) and be in downtown Almaty in 20-25 minutes.
Contact them by e-mail and they will respond. They also rent rooms for half-day. More information here:
http://en.alaport.com/business/hotel.html