Arduino LED 9 V transistor with a common ground
It seems that when you were assembling your circuit, you forgot to add the transistors Q1-Q9 in your circuit. These are needed to allow the Arduino to control when the LEDs turn on/off without supplying them directly, by wiring them to the transistor base, hence you won't need to worry about how much current is available from the Arduino pin. 

If you have access to the GND wires from each column of LEDs, this is where you should place the transistor. 

Once you have fitted it, you should find that you can control the LEDs with the Arduino pins. 

If making any similar projects in the future and you are following a schematic, read it carefully, and perhaps do some Googling or ask here about any parts of a circuit you are unsure about. It is always good to try and understand how something works before putting it together, as that way if you run into problems, it makes fault finding a lot easier.

If you have just a single common GND, then you will need to have a high side switch. You can use a PNP/NPN pair to do something like this. I have simulated the circuit for you so you can see how to connect it. This is probably the easiest, and cheapest way to resolve your issue.

To have the LEDs OFF: 



And to turn them ON:



The logic pin (that switches from 0 to 1) will be a 5V Arduino pin. If you do not have 5V available, it will also work with 3.3V:



This could also be a useful transistor configuration to remember if you ever need to switch a higher voltage for future circuits.