No need to order a car from the hotel to pick you up. Upon arrival, book a car transfer at the counter for airport minibus and limousine services (you will find the counter in the arrivals hall when you exit the bagage area). Fixed (but inflated) fares ranging from 600 -1,200 B depending on your destination. I believe Patong is 800 Bath and Karon/Kata is 1,000-1,200
There will most likely be no public taxi available at that hour (and the saving is not much anyway)