Use a pencil to draw a square around the hole, lining up one side of the square with the center of a joist.
Follow the penciled outline with the drywall saw, cutting along all sides of the square except the one on the joists. Use a utility knife to cut that side. Use a moist rag to dust off the edges of the hole to remove drywall dust and flakes.
Measure the width of the square hole and use the saw to cut the 1-by-4 6 inches longer than your measurement.
Apply the construction adhesive to the top and bottom of the board in a 3-inch wide band, insert the board into the hole and pull it down to the inside of the ceiling, on the side adjacent to the joist.
Apply C-clamps to the ends of the board to hold it in place against the ceiling. Allow the adhesive to dry completely and remove the clamps.
Measure the square and cut a piece of drywall to fit. Use the screws to fasten the drywall piece over the hole, screwing two into the joist, at the ends of the square and two into the 1-by-4.
Cut the mesh tape into strips that are 2 inches longer than the square’s width and apply them to the ceiling, overlapping one another.
Apply the drywall compound with the drywall knife by spreading it over the mesh. Don’t apply too much -- you should just barely see the mesh beneath the layer of compound. Feather the edges to the ceiling. Allow the compound to dry overnight.
Sand the area thoroughly to smooth it and remove rough edges.
Apply a second coat of drywall compound, spreading it an additional 4 inches on all sides, beyond the first application. Feather the edges so that they blend in with the surrounding ceiling.
Allow the compound to dry thoroughly and apply another coat. When this coat dries, sand again.