I think no need to up the car from the monument, but the bus from suvarnabhumi would be better.But there is only one bus company. Name The air-conditioned bus company RUNGRUENG TOUR
The price 305 Thai baht,Buses leave 7.30 am,9.30 am,11.30 am,1.30 pm,3.30 pm,5.30 pm The time is not sure ( But I sure that bus leaves every two hours until 7.30 pm)
Tel +662-132-1888 (No bus number. But is the phone number at the airport. You can ask for more)I'm sorry I don"t know where the bus stop at the airport, I think it's not far.You may have to call more. because there are call center English