The main post office in Bang Rak might be open?
Otherwise maybe ask in a 4/5 star hotel?
Or at the airport?
http://www.suvarnabhumiairport.com/i..._office_en.php
I looked at the Thai Post website, it looks like the rate for a postcard to the U.S. or Mexico, via airmail, is 30 baht ( ~ $1).
Between ~ 11:00 AM and ~ 7:00 PM you should expect to get wet.