How do I get the tire back on the rim

Take the wheel off of the mower. Lay it flat on concrete. Get some vaseline and rub it against the bead of the tire and rim on both sides. Remove the valve core which will let in a lot more air. You need a compressor and not a little one, you need one with a tank on it. You could also use starting fluid or brake cleaner to inflate it but I wouldn't recommend that if you have no experience doing that. If you can't get to air up take it to a tire shop. They will air it up most likely for free.

You best bet is definitely a tire shop, if you don’t have experience mounting tires.