You can get for 60 baht in the air-conditioned S1 bus from Suvarnabhumi Airport (BKK) to the old quarter, backpacker hub Khaosan Road and the Sanam Luang royal field, it takes 45-60 minutes to get there.
Once you exit from the baggage claim hall into the arrivals hall go to floor 1 (BKK has 4 floors) and gate 7, where the S1 bus picks up passengers. Usually, there is staff outside holding a sign that says S1 Khaosan Road. You cross two zebra crossings and should see a yellow-orange US schoolbus alike bus.
Buses run every half hour between 6 am and 8 pm. Visit Transitbangkok Website Bangkok bus line S1. Or watch the video below from the Amazing Thailand Youtube Channel: