8x8x8 multiplexed LED cube with an Arduino Mega 2560

Nice plan!
couple of thoughts:

The board is 180 mm x 120 mm

That's gonna be pricey.
5 boards for $76.20 for example at iteadstudio (2 layer >10cm x 10cm >> 15x20cm price.
Figure out 2 sided & shrink it down. May not be much smaller, my 12xTPIC6B595 board is 100x100mm, not sure of removing the 'arduino' portion and 3 shift registers would let 64 resistors fit instead. Even 10 x 15cm would get you down to $44.20 for 5.
(mine was designed with LED strips on mind, which have their own current limit resistors).

holds all the components (except the Arduino and the LEDs, of course)

So put the 'arduino' on it. Just the 328P, xtal, two 22pf caps, couple of 0.1uF caps, 10K pullup resistor, header to attach USB/Serial adapter to.

Would be a lot easier to review from a schematic.
Add ground plane top & bottom will eliminate a bunch of traces.