it has to be sealed first. Any good primer/sealer will work. Kilz is one brand. Use the oil based one because it works better. The latex one works but it might take three or four coats to cover properly. The choice is yours. All you have to do is cover the water stain and maybe a foot or so past it on all sides.
As for the roller? Use a heavy nap roller, 1/2 inch or longer. The roller full of paint is going to be heavy and messy but it’s the only way to cover popcorn without a sprayer. Just roll slowly or you will splatter the walls, the floor and yourself. A ceiling roller with a drip cover would work too, but just a roller and extension handle, used properly, will get the job done. Or hand brush in several different directions. VERY time consuming. Buy cheap plastic drop cloths for the floor to prevent messy accidents. The last two inches or so near the walls will have to be brush painted. There isn’t any way around it. A roller just can’t get into the corners or near the wall without making a mess.
If you use the oil based primer/sealer, paint the area and then just throw the roller out and wipe the handle down with mineral spirits. Rollers are cheap enough not to bother trying to clean them.
Then you can roll the ceiling in your choice of color with latex paint. If you have a dark ceiling the primer can be tinted to something close to the ceiling color. It makes covering it a lot easier.
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