Here's one way.
Each group of 4 LEDs connected to an Arduino pin represents 1 column of LEDs.
You will drive all the columns, and then drive 1 cathode low via NPN transistor briefly to light up 1 layer. Turn the transistor off.
Repeat for the next 3 layers.
Do it blink-without-delay style, and be preparing the next set of data to be displayed whiletime is passing for the next layer to be written & turned on.
The Anodes can also be driven from 2 shift registers, and the cathode transistors replaced with a transistor array part like ULN2003.
The tricky part of course is the building of the cube.
