The 58 H is the load rating of the tyre so it's quite important to get a similar tyre, click on this link and all the numbers will be explained.

https://lifeatlean.com/motorcycle-ty...les-explained/

Most road/dual purpose tyres are tubeless but because of the spoke arrangement on most rims a tube will need to be fitted. As for whether the tyre fitter will repair your puncture depends on how bad the tyre is. He probably will not repair the tyre as it's the tube that holds the air, so unless it's bald or has splits there's a good chance he will refit the tyre.

You could always learn how to fix the puncture yourself, there are probably lots of tutorials on you tube and quite satisfying once you get the hang of it. You can buy and fit tyres, repair punctures as and when you want.

If you go down this route always keep a spare front and rear tube at home and always fit a new tube, they're about �10 so it's false economy to repair a tube as you can never rely on a patch.

You can do repairs at the side of the road but you need to carry tools to remove the wheel, a bead breaker, tyre levers and a pump to do it easily so unless you're going into really remote areas I wouldn't bother just get the bike recovered home and fix it at your leisure.