Don't try this with a standard RAMPS: it will physically fit, but as the Arduino Due is 3.3V and RAMPS is 5V, it
may damage the processor (by overvoltage at its I/O pins) and/or the heater control MOSFETs (by only partly turning them on so they overheat).
Changing the design to run at 3.3V shouldn't be too hard: the A4988s will work at that, and the MOSFET problem is easily solvable. You'd also need ARM firmware:
this exists, but I don't know how compatible the different versions of ARM are.