The best bottling bucket for Mr. Beer is ... another Mr. Beer.
You can also find El Cheapo plastic ice tea containers at Wally World or Target, which would probably do the trick for ya! (Cool your boiled sugar before you put it in a cheapo container.)
Two ways to fill a container with CO2: if you have a gas bottle, then you probably already know how. If you don't have a CO2 bottle, then the easiest way is a few grams of dry ice. How much dry ice? 44 grams ~ 22L of volume.
Don't seal an airtight container with dry ice inside of it -- people have lost fingers and eyes doing that crap.