How to make a 9x9 RGB LED Matrix

What do you have for an arduino? That will partially drive the answer you get back.

9x9 RGB is really 9 x 27, yes?

So you could have 27 outputs that the arduino drives directly (thru current limit resistors) into RGB Anodes,
and then 9 transistors sinking the common cathodes from each row (up to 27 * 20mA = 540mA).
If you don't have a Mega with 36 IO pins for that, than a few shift registers could be used instead.

Here's the start of such a thing.