You cannot supply 5V at the 2.1mm jack, as that voltage needs to be at least 7V or so to get 5V power to the Arduino.
Additionally, you cannot demand 1.28A from the 5V output of the Arduino as that exceeds the Arduino's voltage regulator capabilities.
The most-likely-to-work approach is to get a 7V 2A wall-wart provided to the 2.1mm jack, then use the Vin output to power your cube. Even that is stressful to the Arduino since this current has to go through a silicon diode (in series with the 2.1mm jack) and when it conducts 1.28A it could get quite hot (and may even be destroyed). Of course, you will have to regulate the Vin output to 5V somewhere else in the cube.
Even better would be to not use the 74HC595's for directly driving LED's, but instead using transistors at the output of each 74HC595. You could then use the Vcc output of the Arduino to power the 74HC595's, but use whatever power source you decide on to power the LED's through the transistors, keeping high currents away from the Arduino altogether.
The Ruggeduino: compatible with Arduino UNO, 24V operation, all I/O's fused and protected