No, you can't connect it the way it is drawn. The pin of the 74HC595 will be exposed to nearly 12V. Either the part will be damaged by the excessive voltage or more likely, the clamp circuit in the IC will limit the pin voltage to about 8-9 Volts, which will keep the PNP transistor on, even when you want it to be off.
Do you really need to switch both the high side and the low side of the LEDs? Connect the resistor to 12V and when the ULN2803 output is on, the LEDs will be on.
If you need to switch the high side, you either need an IC that is tolerant of 12V or you need a level shifter. The simplest is an NPN transistor and a couple of resistors. In this example the "YELLOW LED" is the connection to the level shifter, The 100K resistors should be smaller (1K or so), RTS is the 12V supply and RXD is the connection point for the resistor to your PNP.
Finally You won't get 80mA with this design. 270 Ohms in series with 4 LEDs with 12V supply will be about 30 mA, and it will be less if the LEDs have high voltage drop (e.g. high brightness or white LEDs).