Hi,
One of the (main) features of a MAX7219 is that it drives the LED segments via (adjustable) constant current sources, so it seems a little "strange" to not use that feature. But I guess it has other features that are useful to you.
To answer your question, yes that arrangement of PNPs (as "emitter followers") should work alright provided that the LED supply rail is similar to (not much more) than the MAX7219 supply rail, and the LEDs have series resistors (to limit the current), either in series with, or built into, the LED strip. Also, R13 - R16 are unnecessary because the MAX7219 pulls up the cathode driver pins with 2 mA anyway.
But what are the PNPs supposed to achieve (or more specifically, how are the segments driven)? The maximum segment drive (source) current appears to be 40 mA (typical) and the digit drivers can sink 320 ma (i.e. 8 segments), which seem comparable figures. If anything, for 4 digits, it appears that it is the LED anode drivers that would be the limiting factor.
Cheers, Alan.
EDIT: Ah, I now see that you are driving the LEDs from 12 volts, so NO it will NOT work.