There are 2 bus stations on the Thai side. The official Aranyaprathet bus station some 6 kms from the border. You need a lift by tutktuk or motorbike. This will cost 60-80 baht. You might be able to share a tuktuk and save a bit.
Much easier and cheaper is the other option. Exit the Thai checkpoint and turn right, follow the street for about 100 meter to the market. On the right hand side you will see buses. Some go directly to BKK others dont. Still that makes little difference as long as the bus goes at least to Kabinburi bus station. Kabinburi is a major cross road and you can find transport to BKK anytime during day time.
One bus line which is, unfortunately, at times offering only van services goes to directly to the airport bus station and then on to Ekkamai, the eastern bus station in BKK. From there you can take a free shuttle to the terminal where you can catch your flight or a train into the city.
This link will help you: http://www.suvarnabhumiairport.com/en/118-bus-bor-khor-sor
You want bus line 390. If not available bus 825 from Kabinburi might an option. Unfortunately, I do not have the bus times at hand.
I once took the 3.30 pm bus to Chaosengsao which got me there by around 6.30-7 PM. An airport bound van was waiting.
I guess the numerous options are a bit confusing. HOwever, they show that you do not need to get into BKK and then back-track to the airport.
Thai immigration: You could look up a hotel/guesthouse address and fill this in your immigration form. They usually ask as they need a completed form. Nobody will check this.