Need more hardware, ULN2803 is 500 mA maximum, 32 x 20 = 640 mA.
Same with 74HC595, 8 x 20 = 160 mA, what is more than 50-70 mA maximum current.
I'd suggest to use 4 ULN2803 just as buffers of 595, and 8 mosfets to do a matrix.
That's one of the ideas. My was to use multiplexing. One led at a time with 2ms on/off periods.
1. Pin 10 on hc595 is really named 'SCL'? If so, should you power it there?
2. You can move the resistors onto the ULN and save you some resistors / soldering.
3. If you have pins to spare, or want speed, you could use two sets of sck/mosi pins: one for the hc595 connected to the uln and another for the other hc595s. You can load up the data to serially to the uln, and the one pulse to move to the next column / row. The advantage is that it is very fast. The disadvantage is that it requires a faster refresh rate.
1. I used Eagle PCB libraries, and there pin10 is named as SCL. Should be MR actually. But either way it needs to be powered, and it is.
2. No need for resistors on ULN. It's purpose is to sink current only.
3. I saw this idea on internet and I liked it.

It doesn't mean that at one point I'd like to change it. But at this time I'm not ready for that. I'm a beginner at electronics, this is all new for me and it took me some time to figure it out. So, for now, I'm sticking to it.

Doesn't appear that pins 16/8 (Vcc/Gnd) are shown.
"SCL" is the Serial Clear input. Pulled high as it should be when not used.
Disagree about the resistors - the intent appears to be to drive all anodes, and sink 1 cathode at a time, thus each anode needs to be current limited.
Ideally to about 8-9mA to avoid overstressing the 74HC595/70mA total current sourcing.
Which would limit current into the ULN2803 as well, 32x8mA = 256mA., 32*9 = 288mA.
On the ULN2803, Pin 10 does not need to be connected, that's only needed if a motor was hooked up & the diode was needed.
In eagle PCB there are no Vcc/Gnd pins. It's kinda understood. O.o
One question though. "32x8mA = 256mA." You're counting this as all the anodes are turned on, and one cathode/ULN pin is turned on. What happens when all pins on ULN are on? If I understand well, max current through ULN is 500mA, but NOT per pin.
That's why I think this must be done with multiplexing...
Right?This may works, at least at night time. 8 mA current in 8-column matrix configuration is just 1 mA constant current. Visual perception isn't linear, nevertheless too low. I was thinking about opposite solution, using 4x ULN2803 to INCREASE a current up to 2x (40 mA) or so, make display clearly visible in broad day light...
Been thinking about it. Since this is a prototype, I'll be satisfied with whatever visibility I get.
But, since you got a point, maybe I can edit it now. I've got plenty of time.
What do you suggest? Using ULN on every hc595 with additional power supply? ( 5V and more current ? )