From what I can gather looking at the source for the Flex components, this is not a style that is settable using a normal style property. However, there is a way to get at the button.
If you look at the code for a combo box, you'll see that it's based on the ComboBase class. Inside this class is where the button for the drop down really lives, an mx_internal variable called: downArrowButton:Button. To access it, you can create a class that subclasses the Combo Box class, (you might have to put a "using namespace mx_internal;" in the import section of your class), then access the button directly as:
From there, you should be able to do whatever you want to with the button's style properties. I hope that helps!