Hi and welcome.
That is a single digit display. There is really no need or point to multiplex it. Multiplexing is something you would do with multi-digit displays.
So i think your question is how to drive it with the shift register?
You need to connect the data, latch and clock lines to 3 arduino digital pins. The 8 outputs should be connected to the display's anodes using 8 series resistors (around 200~300R). The output enable pin can be connected to either ground, or if you like, to an Arduino pwm pin, so you can change the brightness. You should also place a 0.1uF cap accross the 5V and ground pins of the chip, close to those pins.
Have a look at the shiftOut() tutorial on the Arduino website. However, there is an error in the circuit diagrams on that page. The 0.1uaf cap should be connected as i described above and not connected to the latch pin at all!