IMO no need to take the overpriced AOT limo service. The cheapest option would be to take a taxi/cab. At least from the hotel to the airport it would also be the easiest option. When you enter the taxi insist that the driver uses the meter rather than giving you fixed fare.
Upon upon arriaval, you need to find the taxi stand on level 2 (not difficult at all). The fare to city should around 300 THB + 50 THB airport fee + high way tolls 65 THB (the latter to be handed over to the driver during driving)
There is plenty on info in this forum about taking taxi from the airport.