The effect in battle for Screen Cleaner states:
Nullifies the effects of Aurora Veil, Light Screen, and Reflect for both sides of the battlefield when the Pokemon enters the battlefield
So if you use one of the aforementioned moves before Mr. Rime enters, then damage will be applied with the reduction of what the move offers. However when Mr. Rime does enter the battlefield then the effects will be removed from both sides, assuming both sides placed up barriers that it. To best use this effect try to predict if your opponent is running screen type moves. Look for Pokemon that are known for running such moves and plan accordingly. I do not advise you to use him if you're planning on using the aforementioned moves yourself. It would not only hinder your chance of winning but also make running the moves pointless if you're going to nullify them.