Well, it depends....! In many older Cypriot buildings - even quite prestigious ones, such as the Holiday Inn in Nicosia - you can't flush paper. This is because they were built with a foul-water arrangement that, because of its design, eventually blocks up with paper. On the other hand, lots of newer buildings have conventional European arrangements so there's no problem, and a lot of older buildings are being converted (it's not difficult or expensive to do, and in our 150-year old stone village house you can flush away to your heart's content!). Generally speaking, there are notices telling you if you can't; if there are no notices, you probably can.
A bigger problem at the moment is the water shortages, so that in many cases you can't flush anything! Up here in the mountains we're OK for water (touch wood!), but in the resorts and in Nicosia the supply is severely restricted - only three or four hours a day in some places.
As scoob says, however - don't let it put you off - our island is beautiful and you'll get a warm welcome.