Repairing water damage is never a pleasant prospect, but drywall damage repair is less onerous than repairs to some other materials. Ceiling water damage repair for drywall is rather more complicated than repairs to walls, and presents some extra issues.
First of all is the possibility that enough water may have accumulated behind the ceiling drywall that it presents a danger of collapse. While most modern houses have 5/8 "fire-stop drywall on the ceilings, this can be a blessing or a curse. It is a blessing in that the extra dimension of the material will withstand greater water weight before collapsing catastrophically. On the other hand, if one is not able to safely drain out that water before collapse, not only will there be a greater weight and volume of water accumulated, but the extra heavy drywall material itself can present an increased risk of injury to people and things beneath .
Once the source of the ceiling water damage is accounted for, if there is a downward bulge in the ceiling drywall, you will know that there is some amount of accumulated water suspended there. After removing all furnishings and other moveables from the affected area, lay down plastic sheeting to catch the inevitable falling of water and soaked drywall. Obtain a supply of buckets and ideally, a wet / dry vacuum and prepare to catch as much of the water as possible. Pierce the drywall at its lowest point and catch the outflow with the vacuum or in buckets. Once you start, the water will flow until it's gone, so do not skimp on the buckets.
Once the immediate danger of collapse is removed, you can proceed with drywall repairs as in repairing wall water damage. However, note that working overhead presents problems not associated with repairing walls.
Drywall is heavy, and the thicker, fire-stop types are the heaviest of all. If you have to replace any area of substantial size, plan to have one or more helpers to hold the patch in place while it's secured with screws or nails.
Also, take account of the fact that you'll be working overhead, and some means of reaching the work area will be required. Doing this sort of work can be accomplished on a ladder. However, be aware that working on a ladder on a patch that you can not reach all of will require trip after trip up and down the ladder. Drywall repairs demand a number of steps, all of which must be done in their own time. A good idea is to buy or rent scaffolding to maximize the amount of area you can work on each trip up and down. It's safer in the long run, and your knees will thank you when it's over.