Vacuum the popcorn-textured ceilings to remove trapped dust and debris. This may improve the ceilings to a point where you don't feel that they need painting anymore.
Move all the items in the room you want to paint outside.
Hold plastic sheeting to the top of the wall where it meets the ceiling, then tape it in place with painter's tape. Cover the floor with drop cloths. This protects wall and floor surfaces from paint.
Wear protective clothes, such as long pants, a long-sleeved shirt and a hat, to prevent the paint from getting on your body.
Fill the airless sprayer tank with an alkyd primer. Don't use water-based primer because water may cause the texture to come loose from the ceilings. Hold the gun of the sprayer about 12 inches away from the ceiling and pull the trigger, then move the gun until cover the entire ceiling with primer. Let the primer dry overnight.
Fill the airless sprayer tank with flat latex paint, then apply the paint using the same technique. You may have to apply two or three coats of paint to achieve good coverage.