Fixing a hole in a gas tank can be done securely with basic household tools and a little elbow grease. Making it a bit more DIY friendly is the fact that virtually all modern riding mower tanks are made of plastic, so there is no need for soldering or welding like there would be on a regular metal type of gas tank.
Remove the tank from the riding mower by using your screwdriver to unscrew the screws holding it on in a counter-clockwise rotation. If there is a hole in the tank, the chances are that any fuel has already leaked out, but the tank can be drained by removing the fuel hose clamp with a screwdriver and then draining the remainder of the gas into a regular gas can for reuse.
Wipe off the outside of the gas tank with an old rag, getting as much dirt and grit off as possible. It doesn't need to be spotless, just wiped off enough to remove any foreign objects.
Sand down the area to be repaired with your 100 grit sandpaper. It does not have to be sanded smooth, just sanded enough to be roughed up. The ridges from roughing up the plastic will allow the adhesive to settle on and grab hold, making for a much stronger bond.
Cut a piece of screen material to cover the hole and overlap the sides by about 1/4 of an inch. This doesn't have to be exact, but there absolutely needs to be a bit of overlap.
Apply the plastic epoxy adhesive to the tank over and around the hole. Extend the adhesive well and away farther out than the size of the screen piece you have cut.
Press the material piece onto the adhesive that covers the hole. Make sure it is pressed firmly into the adhesive and that all sides and corners are pressed into the glue.
Cover the entire patch with more epoxy plastic adhesive, and don't go lightly. Make sure the entire material patch is completely covered, and also make sure that the adhesive is well applied past the ends of the patch. When this is done properly, there should be no sign of the patch showing through.
Allow the epoxy to dry per the manufacturer's instructions. Some epoxy formulas cure within five minutes, and some will take hours, so choose the type you are most comfortable working with.
Remount the gas tank, attach the fuel line and fill with gasoline.