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Rough sketch of a 4x4x4 LED cube. Remaining Arduino pins should be connected to the remaining LED Anodes. Not illustrated are three more layers of 4x4 LEDs. Anodes of those should be connected to the Arduino (shared with the 16 LEDs in first layer). The Cathodes should be tied together and connected to the collectors of the remaining transistors. A0 thru A6 should be set as digital outputs using pinMode command.
Arduino sketch should setup the Anode pins in the desired 4x4 pattern for layer 1, then turn on the transistor for that layer. Repeat this for the other layers as quick as possible for the illusion that all 64 LEDs are individually controlled.
Disconnect the Emitters from ground (using that switch) during sketch upload from Arduino IDE, otherwise D0 and D1 might be pulled to ground and disrupting the upload.