There is a sweep spot you can get protected by the bubbles (which are your main concern) right below the first rock.
So basically you have to dodge his bullets and then jump in the moment where he raises the platforms, then look for cover below the first rock/platform on the left. When you are safe, jump above and shoot him. Rinse and repeat.
You can see my strategy in the following video:
Sorry, this Game Clip has been removed