I have that reader, reading the Mifare Classic tags. It uses the SPI interface, but I also have another device on that bus. Even though the two should be compatible, sooner or later, the software always hangs up, but if you aren't using two devices on the bus, that shouldn't be the problem.
Also the device is a 3.3 v. device. Have you taken level shifting into account? Are you trying to run it from the 3.3v supply on the Uno? That supply doesn't supply much current.