A ride on mower uses tires that are very similar to those you would find on your car, which means that they are just as susceptible to punctures. Mower tires, however, may suffer flats more often because of all the rocks, sticks and other debris hiding in your grass. If you get your hands on a tire repair kit (available at any good auto store) you can repair the puncture yourself.
First, move the ride on mower to a level area and raise it about 6 inches off the ground with a car jack (either at the front or rear of the mower, depending on which tire is flat). Turn the flat tire until you spot where the puncture has occurred. If there is still an object in the hole, you will need to pull it out with a pair of pliers.
Next, retrieve the reamer tool from your tire repair kit (it looks like a round file with a handle) and insert it into the hole. Push and pull the tool in and out of the hole a few times, as this will create a more even puncture and remove any excess rubber. Remove one of the tar strips from the tire repair kit, pull it off its backing and insert through the end of the threading tool (it looks like the eye of a needle on one end).
Using the threading tool, insert the tar strip into the puncture until about half an inch of the strip remains outside the tire. Give the threading tool a twist and quickly pull it out. The tar strip should be released from the tool and remain in the puncture. Finish by filling the tire with air and taking the ride on mower off the jack. It should be good as new.