Since you are arriving TG F getting out of the airport would be fairly quick and simple. I believe there are left-luggage facilities but I have never used them. Unfortunately at 5:30 there is a lot of hustle and bustle but not a lot of typical tourist attractions are open. One thing you might could do is visit the flower market. But that wouldn't kill a whole lot of time. But from there you could wander around China Town. If you then get to the Grand Palace at the time it opens you could make a quick run through.

Since you are departing EK business class your check in should also be fairly quick. You should check (maybe the EK forum) to see if EK offers fast track departure passes since departure security has been a problem for some folks. Given your schedule you would want to be heading back to the airport by 11. That should give you time for a quick shower (presumably EK has showers in their lounge? something else to find out) before your flight.

I suppose it depends somewhat on how adventurous you are, have you spent much time in crazy Asian cities, etc. It may also depend a bit on how long your flight into BKK is. I imagine it is an overnight from Europe? So even flying F there may be some jet lag. Sitting around the airport for 8 hours doesn't sound like much fun to me. You will probably need to sit around waiting to check in for your EK flight since there is no arrival lounge privilege you'll be out in the airport.