Here is the definitive guide to cleaning a Honda carburetor (.PDF file):
How To Clean A Honda Carburetor
That said, your time is worth something, especially if the equipment that's down isn't making you money. A new carb from Honda runs about $65. Have the serial number from the engine block if you buy one; it will make it easy for the parts guy to find the correct carb quickly. (There are a number of different part numbers for the GX120 engine's carb, depending on which version of the engine you have).
Caveat: I work for Honda, but the preceding is my opinion alone.