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.