The squares on the boxing ring are bigger than the squares on the map, so there is no possible way to line it up properly. How are people handling this? Should I just take a bunch of special terrain markers and glue them to a piece of cardboard to represent it?

Align the lines on the ring as well as you can, looking straight down on the map when drawing lines of fire. The ring itself will stick out a little into the squares surrounding, but the angles do work out.