Do you guys know if there is some commercial cable that allows me to connect the LCD Shield to the Arduino Mega?
It is very unlikely that such a cable exists.
Define far. Ambiguous questions lead to guesswork and wasted time.
Probably the better way is to attach the LCD to another Arduino (cheap Uno clone?) and use a wireless link (NRF24) to send the data form the Mega to the Arduino attached to the display. Long cables pick up noise (especially the analog button signal).