1) I'm afraid there is no direct bus to airport, neither from Moshi, nor from Arusha. Find someone share a taxi might be your best bet, or you can book a taxi alone for $40 from Moshi.
There is local bus stop at the cross of main road and airport road. Not sure if local bus departs that early before 6 AM.
2) If you want to do all of 潜水,晒太阳,参观石头城, one day is not enough. IMHO, walking tour around the stone town worths at least 5 to 6 hours. From waterfront on the west to the main market on the east, there are lots of interesting alleys and doors to see, if you are interested.