How to switch/power a 13x13 LED Matrix?
I would suggest that you should use a shift-register chip that has decent current-sinking capability for the cathodes, and use a shift-register chip to drive discrete transistors for the anodes.  Perhaps wire the matrix as 7x26 and use two TLC5925 chips for the columns, and use a 74HC164 or equivalent to drive seven nice beefy transistors for the rows.

Actually, it may be a good idea to rig up the rows with a counter chip and a 555-timer wired so that they will automatically scan, but the main processor can 'nudge' the timer when it's almost ready for its next count.  Such a circuit could ensure that no matter what the processor did, it would not be possible for a row to be energized much more than 1/5 of the time (the processor could strobe six rows quickly, then linger on the seventh, then six rows quickly, linger on the seventh, etc. but the hardware would limit what fraction of the time would be spent on any one row even in a worst-case situation.