OK, lets take a look at the schematic for this cube.
First of all, I have to say I don't like it, for various reasons. It's over complicated and overloads the components it uses. It does not in fact use shift registers. It uses 8-bit parallel latches, making the wiring much more complicated.
But we can estimate the current requirements:
Lets assume you are using blue leds and their forward voltage is around 3.2V. The schematic shows 100R series resistors for the columns. So the current flowing through each led in a layer would be around (5-3.2)/100 = 18mA. Each 74hc574 chip will need to source 8 * 18 = 144mA (the maximum for that chip is 70mA). There are 8 of those chips, so the total current would be 1.15A.
Assuming the cube is multiplexed by layer, only one layer would be lit at any instant, so the total demand from all 512 leds would still be 1.15A.
In addition, you want your PSU to power the Arduino, and there will be some small current required by the rest of the circuit.
So if I have estimated correctly (Mike, please check my assumptions & maths), your 1A PSU is not adequate. However, because those '574 chips are overloaded, they may in fact limit the current significantly, by dropping a large voltage and getting hot.
Even if you purchase a 1.5A or 2A 7.5V PSU, you will still have a problem. You will be relying on the 5V regulator built in to the Arduino to regulate the 7.5V down to 5V. That regulator is not rated for more than 1A, so it would overload.
I recommend you buy a 2A 5V regulated/switch-mode PSU. This can power the cube and Arduino, You would power the Arduino through its 5V pin instead of Vin pin or the barrel socket.