When you directly connect the USB cable from the Sony Xperia ZR to Xubuntu 13.04 computer, and try to tell it to connect the USB from the phone, it displays a message asking Install PC Companion. If you select Yes, it shows you a folder where you see the PC Companion software, an EXE file that it expects you to install.
This is not what we want to do. We want to access the external SD Card after connecting the phone to the computer. To do that, disconnect the phone from the computer, go to Settings -> Xperia -> Connectivity -> USB Connection Mode -> Change the option from MTP to MSC and then connect the phone. This time, you will find that you are able to mount the SD Card you installed on the phone like it appears on the Samsung Galaxy Y.
Make sure that after you transfer your files, you unmount the file in Linux and then choose Disconnect SD Card on your phone.