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You can carefully vacuum or lightly brush off the worst of the dust so you don't damage the texture. Using any liquid cleaner may cause the ceiling to smear or even lift off, necessitating more repairs.
Regarding painting "popcorn" ceilings, the safest thing is to first coat it with oil primer, even if you think it may have been painted in the past. This will assure a good base for your finish paint. An oil primer will not lift the ceiling material off (unless the ceiling has been exposed to a leak from above) and gives a solid, water resistant surface that can be repainted with any latex of oil finish paint.
I prefer Kilz or one of the low odor, fast drying oil primers... they save lots of time! Let the primer dry for at least a few hours and then coat it with one or two coats of a quality acrylic flat ceiling paint.
In kitchens, there is always the possibility of something splattering on the ceiling. If you want a finish that is not very porous and cleanable by sponging, use kitchen-bathroom paint for your finish coat. A number of companies make them, such as Benjamin Moore and Zinsser. These specialty paints have a low-luster finish that is washable and mildew resistant. Two coats are required for proper sealing. Zinsser sells their kitchen-bathroom paint in two finishes... satin and gloss... to suit your own aesthetic sensibility.
Spray textured ceilings are only "somewhat washable" regardless of the paint used, since you can damage the texture with vigorous scrubbing... as well as your sponge! To clean, spray with a cleaner and sponge off with a daubing action. Obviously, quick action is ideal. If anything dries, get as much off as you can and then just touch up the ceiling with a little of your ceiling paint. Depending on the stain, it may take a few coats or even a dab of primer first to keep the stain hidden. Textured ceilings touch up nicely for a few years.
Sprayed acoustic ceilings are relatively fragile, especially if they have never been painted over. They cannot be washed and the bits of texture have a tendency to fall off. More than one of my clients has complained about texture in their soup!
The absolutely best way is to vacuum the cobwebs off. Cobwebs tend to be rather "greasy" since they catch all sorts of air-borne dust and cooking smoke. I avoid using any sort of dusting rag or cloth because rubbing them onto the ceiling can cause marks that are difficult to remove.
The second best choice would be to use a feather duster, but being careful to lift the cobwebs from the ceiling. Remove the cobwebs from the feather duster before doing any other dusting or they may leave marks elsewhere!