After the LED cube is soldered there should be nine anode leads and three cathode leads (one for each layer). Hook up the anode leads to the breadboard. Each should have its own strip. Connect a 220 kohm resistor each anode and wire to a pin on the arduino uno. Each anode will have it own pin connection on the arduino.
Now connect the three NPN 3904 transistors to the breadboard. The controller and base should be connected to separate strips on the breadboard while the emitters can all be connected to one strip. Connect the emitter strip to ground. Wire a 22 kohm resistor to each base lead and connect to a pin on the arduino. You sholud now have a total of 9 + 3 = 12 pins wired on the arduino. Connect each controller lead to a separate cathode connection on the LED. Each controller should be connected to a different layer.