If you decide to go out of the airport, first check if you can get a visa (depends on nationality). If yes, it cost a few dollars and can be had on arrival, paying by credit card. At that early time in the morning the best thing to do would be to go for a long walk along the Corniche - 7km seafront walk.
A taxi will get you there in less than 10 mins. Early morning temperatures are in the high teens, low 20s (C). The biggest mall is called 'City Centre' and is at the far end of the Corniche from the airport, about 20 mins by taxi, but not sure what time shops open there - it may not be until around 10.
Otherwise if you want early morning shopping, ask for the 'new old Souq' area which is only 10 mins from the airport.