Michiel_:
Note that I've never ever drawn a circuit before, so if there are mistakes, please tell me
Sure! Looks great!
First confusing aspect for those reviewing it is that you have drawn it upside-down - convention is to draw the ground side - almost always the negative - on the bottom and the positive power supply on the top.
Michiel_:
Well, I thought resistors would be unnecessary since an LED can be driven by 3.3 volts...
So you do not understand what a LED is - indeed, it is a diode, with the characteristics of a diode, specifically the "threshold" effect whereby up to a certain voltage it passes essentially no current, and at or around that threshold voltage, it will pass however much current you feed it with rather little increase in voltage and that voltage threshold is dependent on temperature. So a resistor is not used because you have "more voltage than you need", but to determine the current to be the correct value.
Similarly, the base-emitter junction of a transistor is a diode with an approximately 0.7V threshold, so needs again, a series resistor to determine the current.
And I wonder what your "3.3" battery actually is?
Since you are going to power it with an Arduino UNO, you will actually be running it at 5V, so the voltage drop of the transistors will not be a particular problem. Now for multiplexing, you have to decide whether you are going to "strobe" by rows or columns. Whichever grouping you will use for the strobe, selected one after the other, will need to provide the current for four LEDs, so you would want something like 1k resistors in series with the bases of those transistors. The other direction grouping you will use to place the pattern for each strobe, needs 220 ohm resistors in series with the LED connections to the collector, and a 3k3 resistor in series with the base of each of those transistors.
Now actually, with modern efficient LEDs (which I decline to refer to as "super-high-brightness" or such as they are no more or less than efficient in comparison to early model LEDs which were simply awfully poor - I essentially need to throw out my stock of the older LEDs as the are simply too poor to consider using for any project ), 20 mA even for a quarter of the time as they are strobed one-in-four, they will be very bright so in fact, you could just as well use 1k resistors in series with the side of the array that is not the "strobe" lines, and use no transistors.