It might fill the tire and make it round but it probably won't hold much weight. If it does hold weight it would probably be a rough ride to push that thing around anything that isn't smooth ground. A tire patch like @darkwing_duck suggested or a plug would probably give better results.
Do It Yourself
Make it, Fix it, Renovate it, Rehabilitate it - as long as you’ve done some part of it yourself, share!
Especially for gardening related or specific do-it-yourself projects, see also the Nature and Gardening community. For more creative-minded projects, see also the Creative community.
This community's icon was made by Aaron Schneider, under the CC-BY-NC-SA 4.0 license.
I had the exact same thing happen to me (previous owner left old wheelbarrow). And what I did was take the wheel off, pop the tire out and put a bicycle patch on the hole in the tube. Good as new.
That being said I put probably 25+ bicycle tire patches on in my life, so it was cake for me.
I'd recommend just going for a new wheel and tire. Harbor Freight or your local equivalent probably has one that will fit with a solid tire under $20 and never having to worry about flats or airing it up is worth it.
There's also a solid chance the tire is really old and doesn't have much life left anyway. By the time you patch it a couple times, you've probably wasted more than $20 of your time.