Drywall installation on the ceiling without help
Get a 12' 2x4 and a 3' to 4' 1x4. Screw the 1x4 at right angles to the 2x4 near one end. The 1x4 should be centered lengthwise on the 2x4 and overhang the end about 1/2". See sketch below. In the sketch I showed screwing the Tee board on the edge of the 2x4 but you may want to screw it to the flat of the 2x4 to get a better screwing to make the 1x4 fairly solid. Use 4 or 5 screws for this. 



Cut the total length of the 2x4 to be about 3 to 4 inches longer than the floor to ceiling height. Now you climb the ladder and hold the drywall in place and have another helper hoist the 2x4 up with the cross piece coming up under the piece of drywall. The helper should be able to wedge the 2x4 in place to hold up the drywall until it can be screwed into place. If the wedge wants to kick out on the bottom then it may be necessary to shorten the 2x4 a slight amount. 

The power drill you can purchased can be used for installing drywall screws but it can take practice to avoid sinking the screws too far or even all the way through the drywall. I would recommend instead using a hand screwdriver for this small job. Make sure to use proper drywall screws. A common size for this will be 1 5/8" length screws. When installing drywall screws you do not pre-drill the drywall itself. The screws are also very sharp and will not need any pre-drilling into the framing members either. When setting the screws use care to tighten them down just enough so that they dimple the paper and recess a small amount. You do not want to tighten so far that the outer paper layer tears.