You can certainly use your Test URL to test the upgrade process. However, keep in mind that it can only be used privately, public access is not allowed.
Additionally, if you keep your Live URL open and perform the upgrade on your duplicated live instance on your Test URL then repoint your domain, you will lose any data/content/etc... made on the Live URL during that time. You also cannot point two different software versions at the same database as it would break. The best move is to a test upgrade on your Test URL till you're comfortable and then shut down your Live URL and perform the upgrade on the Live URL. That will minimize your downtime as much as possible.