Rinse your side feed airbrush between colors. This prevents excessive paint build up and paint colors mixing with one another.
This method, also known as a color change rinse, is the best thing you can do to prevent future airbrush problems before they have a chance to become problems.
Do you need to clean the paint bottle too? You might not actually have to clean that bottle. Just a thought.
If you need to clean the cup, drop it into a jar or your cleaning solution and let it stay there for a few minutes. Then have at it with brushes.
Do you have a screw in nozzle?
Do you have a self centering nozzle?