That screen is not a normal screen. It doesnt work on normal screen plugs but as you say a aux-1 as defined on ramps board.
It requires access to the serial ports of the controller.
If you look at the 10 pin EXT plug on the melzi It can be made to work
You need to set the screen for external power, as the melzi cant supply enough.
then connect up gnd, D10/RX1 and D11/TX1 from EXT to the display
Then change your firmware to user serial 1
ie #define SERIAL_PORT 0 becomes #define SERIAL_PORT 1 in configuration.h
this will stop your usb cable from working and enable the screen
If the baud rates are the same, it should now work.