If it's a puncture, you can plug it. Walmart, auto stores, and hardware stores all carry tire plug kits. It's a piece of rubber typically covered in a sticky paste that you insert into the puncture with a simple tool. Be sure to clean the hole first with a file tool (should also come with the kit, and is sometimes on the shaft of the plug tool as a 2-in-1). It's simple and is something you can use on any tire without a tube. If you have a "fancy" wheel barrow that has bicycle type tires (like some of the newer multi-purpose models on the market), your would treat it like a bicycle and change or patch the innertube. There are glue on rubber patch kits available anywhere that sells bikes as well as hardware stores.
As for the leak-proof tires (solid foam filled tires that do no rely on air pressure), they can take a beating but I really don't care for them. In my experience they are stiff and don't perform well under heavy loads and over ridges/bumps like curbs or the edge of boards.
