Be very careful with those LEDs, they run at 10mA (or less) and have a max current of only 11.5mA. Most circuits on the web assume 20mA LEDs.
Each LED needs 4V so that circuit won't work as-is.
WITH MODIFICATION, The "2N3053" part of the circuit is a good way to do it but use 19V instead of 6V, this allows more LEDs. (nb. Many laptop power supplies are 18.5V/19V - try and find one).
Put 4 LEDs per column instead of 2, use 220 Ohm resistors instead of 100 Ohm.
Put 11 strings of LEDs per transistor instead of 10, that gives 44 LEDs per transistor.
132 LEDs can be done with three transistors (44 x 3 = 132).
You can connect three transistors to a single Arduino pin if you use 470 Ohm resistors between the Arduino and the transistor (instead of 220).