Most likely priming and painting will take care of the problem. "Prime it" refers to using a special coating, primer, to seal the surface and get it ready for painting. There are many types but a good interior acrylic primer will work well.
I don't know of a test to determine exactly what the dust is. You might be able to find a lab locally in the phone book. Most likely this testing will be expensive.
One easy test you can do is wipe the ceiling with a clean rag and warm water to see what comes off. A good cleaning might be needed before priming and doing this yourself would save you money and let you know just how much dust is on the ceiling.
If the rag is colored the same as the ceiling paint then it might be paint. If large areas come loose from the water then something else is going on and painting might not address this new development.