Sure. Just log onto the site with an account on the internet with a username and password and you should be good to go.
This can be done on the site or the App, then just pick Tourney Mode.
Do note the screen will tell you when the Tourney will start and when you can compete for prizes. Regardless, the Tourney Room is open for fights at any time.
I had a similar problem with a friend's account when I introduced him to UR. Apparently once connected to a facebook account, it won't work or automatically redirects you to facebook UR. That's why I never connected this account to a facebook account. There may be a way to desync the two.
Well the other way to play is go to the bottom of the page and download the beta, just unzip it and play it on your desktop nice and easy but make sure you have an account becuse you can simply login with facebook there