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.
The best example I can think of is the "Peggy," A Light Emitting Pegboard Display. It is a 25x25 LED matrix display driven by an ATmega168 (which is pin compatible with the ATmega328)
The wiki page has a lot of good information. Including a detailed schematic.
There are few things to notice in their layout.
For one, they use a row common anode setup. That is the current source is on the row, and sink on the column. You have yours in row common cathode. There is nothing inherently right or wrong with either layout. Just something to keep in mind when designing your circuit. If using discrete leds, it just means flipping the led connections. If using a prebuilt LED matrix, it is something important to know. (I'll assume you can easily swap the order to match the peggy schematic. If not, just swap column for row in your head)
They use 74HC154 4-16 decoder/demux chips for row select. Since you only need 10 rows (or cols) you can get away with just one. Of course, there is the issue of current. In your case, at 10 x 30mA = 300mA minimum. To solve that problem they used 2STX2220 PNP transistors which will be able to source up to 1.5A per row. A bit over kill in your case. Since you will just use these as row select switches, just about any other pnp transistor that can source your max current should work just as well. Take a look at Transistor Circuits to figure out what resistor values you'll need for full on/off operations.
On the Peggy board, for the column sink driver they use an STP16DP05. But I have found these difficult to find and expensive. There are many other alternatives like the TLC5916 These use a serial input, and can be easily cascaded. If not, a digikey of mouser search for led sink driver will yield many results.
Alternatively, since you already have ULN2803 arrays, you could use two of these with a single current limit resistor per column. That's a lot of pins, so you'll have to get creative, but it could work for the column sink as well.
Avago published a nice application note titled "Introduction to Driving LED Matrices". It covers this and a few other things.
Best Answer
The 2N3819 is an RF 'N' channel depletion mode JFET and is not really suitable for LED switching. Depletion mode means that the gate needs to be driven abut 7 volts more negative than the source to turn it off, and driven to the source voltage but no higher to turn it on. The maximum drain current is probably not high enough for your needs either.
Look for an N channel enhacement mode MOSFET instead.