Your choice of cable may not work due to the fact that card reader uses MAX232 which is a 5V device (it’s input /output signals are TTL 5V) while the PL2303 in the USB-Serial cable is a 3.3V device. This may cause malfunction. To overcome this issue, you may use MAX3232 which is 3.3V device.
Another big reason for this cable to not work is that it only provides you with TX and RX while your circuit also needs RTS signal which goes to the RESET of the memory card. The only solution to this problem is to look for some other cable which also translates hand shaking signals.
You should try to find a USB-Serial cable which has TTL 5V signals and which translates at least RTS along with TX and RX.