I have written a post about how to go to Bangkok city center from Suvarnabhumi Airport which tells you the transportation modes that you can use to get to the city center of Bangkok from the airport.
In short, you can use airport limousine, taxi, airport rail link, bus, and van to get to Bangkok downtown. Unfortunately, most of the services are not available 24 hours a day. Airport rail link and bus (including BTS Skytrain and MRT Subway) closed around midnight and starts again around 5.00-6.00. So, if you hit Bangkok after midnight, you would have only 2 choices which are airport limousine and taxi.
And to save your money, taxi is the best option. The taxi fare in Thailand is considered cheap when comparing to other big cities. You can see the taxi fare rate in Thailand here. Well, let’s say around 200-500 baht, you can reach the city center of Bangkok.
The good thing is no traffic jam at night. You can reach your hotel faster and cheaper than during the day. To save more money, tell your taxi drivers that you do not want to use expressway.
All Thai Taxi now are using the meter. The only problem is a taxi driver may refuse to turn on the meter and offers the flat rate fare which is more expensive. My recommendation is to not go with that taxi and try the next one. Wish you luck!