Do what the locals do-taxi from airport

before you join the custom clearance Q and get your immigration things done, look out for some maps. Personally prefer the maps obtained here as they are more detailed(and drawn to scale) than what the concierge provides.

When you arrive at the arrival hall, look for the sign of ''departure hall''. Look for the escalator and go up to the floor. Take your taxi from there!

Just say ''hotel name, bai meter''. Bring with you your booking receipt from your hotel. Usually this also contains the address in Thai language. If the driver look confused after you told him your destination, just how him the address. Remain calm and tell him to go by the expressway, which they usually refer to as the tollway. You might have to 'pay as you go' , ie paying for the toll charges whenever you reach a toll gate. It is 20B per gate, and there were 2 gates before you reach downtown. so get 2x 20B ready!

How much does this cost? 250-300+B, after tipping!

Oh yes.. the driver understands your "by meter"... The Thais say "bai meter" for it anyway.

but... if money is not a problem.. just save the hassles. Just "order" an Bangkok airport pick up service through the internet or your hotel.