Thanks for so many replies, they are really helpful.
Boffin1, could you post the schematic you mentioned? I'm still learning so even if I won't use it it will be a good lesson for me.
I'm still curious to know something about this method:
CrossRoads:
This design will control 64 LEDs. You need 72. So what you can is add additional banks of LEDs, and control additional transisitors with a shift register instead. Or a shift register with high current sinking capability, such as TPIC6B595, and skip the transistors.
Take it a step further, and drive the anodes from 2 shift registers, and free up IO pins for buttons for setting time, retrieving time from an Real Time Clock chip, etc.
I want to make a clock, so I'll need two independent circuits - one for hours and one for minutes. On the schematic you've posted I see that all arduino pins are used, so two led circuits will need 2 arduinos, right?
I can lose half of minute diodes (one diode will mean two minutes) and rest of free pins will be used to hours circuit. Or maybe multiplexer - what do you think?
The other way is using chips, as some people posted here - I have really zero experience in electronics - I will just have to solder those chips into arduino proto pcb?