If your idea is just to patch the area where the plaster got soaked and fell off, here's what I'd do:
Buy a sheet of drywall (also called plasterboard or drywall). It comes in 4'x8' sheets. You'll want the 1/2 thickness for a ceiling so that it doesn't sag. Drywall is very inexpensive (about $5-8 a sheet). Also buy a small bucket of pre-mixed drywall compound, and a small box of 2" drywall screws. Probably run about another $10-12. If you have a few more dollars in the budget, purchase a roll of paper drywall tape (about another $5).
Other tools you hopefully have (or can borrow) are: an electric OR battery operated drill (12 volt min. on the battery), a phillips head screwdriver bit for the drill, a 4" wide putty knife in very good condition (or some actual drywall taping knives), a utility knive (the kind that use heavy-duty razor blades), a drywall saw or an old cross-cut handsaw (optional), a ladder, and some drop clothes to save the floor from wet drywall compound (sheets will do).
Remove the damaged plaster and try to get back to edges that are tightly stuck to the ceiling. Be careful, because old plaster is very brittle and you may end up pulling down stuff that just moments before was perfectly good! Try and make the hole as square as you can (only because it will be easier to match up with the drywall patch... it doesn't have to be perfect or completely square).
Leave the lathing strips in place (those long strips of 1/4" thick wood that are running underneath the plaster). You can put the drywall right over them.
Cut the drywall to fit using a utility knife (if you need addtional information about how to do this, let us know.) You can also use a drywall say or an old cross-cut hand saw if necessary.
Mark the ceiling around the edge of the hole where the joists run across the hole. This will help you later when you need to screw the plasterboard to the ceiling and can no longer see where they are. (Missing the supporting joist with the screw leaves a hole that must be filled, and PLASTERBOARD IS HEAVY so having an exact idea of where you can run the screws and have them hold is time-saving.)
When you position the drywall patch in the hole, you may find that the drywall is not as thick as plaster. Use strips of compressed cardboard (the kind of cardboard found on the backs of tear-off paper pads... not the corrugated kind used to make boxes) to build up the thickness of the drywall until it's even with the paster at the edges. You can use masking or duct tape to hold the cardboard to the back of the drywall.
Screw the drywall into the ceiling joists. Be sure to set the heads of the screws JUST beneath the surface of the drywall (it should be just a slight dimple... not a hole).
Using the putty knife and the drywall compound, put down a generous layer of the compound over the surface of the crack where patch and ceiling meet. If you bought the drywall tape, now is the time to bed strips of the tape into the compound, then put more compound over the tape. Smooth this first layer, but don't fuss too much. Get it as level as possible... but it doesn't have to be perfect at this point. Just don't leave big glops.
Also put compound over any of the remaining screw heads.
After 24 hours, use the putty knive (or a larger drywall blade) to apply more compound, and smooth (try to feather the edges of the compound onto the surrounding surfaces so there's no obvious edge).
After ANOTHER 24 hours, put on one more layer of coupound. At this point, it should be looking pretty smooth.
When all is dry, you may have to sand the compound slightly to remove any obvious blemishes, but now you just need to prime and paint. Should be good for another 50 years! Have fun!