"In terms of the 595, I am only just exploring this myself so I can't really offer any advice on that yet, but with that display (4 Anodes, 7 Cathodes) you can control the entire display from the Arduino to get the multiplexing sorted before adding the extra complexity of a shift register?"
No. You can use the arduino to source current to segments (with a current limit resistor per segment) but you need a ULN2003/2803 type part, or 4 transistors, to sink the combined 140mA that could go thru each segment. The Arduino cannot sink that much current.
Or, limit the current sourced to each segment to 20/7mA, ~3 to 4mA, then an Arduino pin can sink the common cathode current.