Hi Justin,
Your pics must "sequence" through the 23 columns of LEDs quick enough that they don't appear to flicker, about 50Hz at the slowest but much quicker would look best. Yes, the LEDs light and turn-off instantly.
Since each LED will be accessed for only 1/23 or less of the total time, its average current (and brightness) will also be only 1/23 or less of its peak current.
With your 2N2222 column driver transistors at their max current rating of 500mA, a lit column of 7 LEDs will each have a peak current of 71mA. Therefore the average LED current is only 71/23= 3mA, which is fairly dim.
LEDs that are made to be multiplexed have a peak current rating of 1A or more. Then with suitable high-current row and column transistors and a peak current for each LED of 460mA, their average current will be 20mA and they will appear bright.
To determine if your LEDs are suitable for your application, check their datasheet:
1) What is their angle?
2) What is their max peak current rating?
3) What is their max continuous current rating?
4) Is their brightness linear with increasing peak current, or does their brightness "flatten-out" (saturation) at high peak currents?
5) What is their typical forward voltage at a high peak current?