It should be fairly easy in that it is similar to what you already have. The LED connections for the resistor and power side are identical to what is there now. The change us that the wire from the bottom of the led that currently goes to the arduino now goes to one of the pins on the shift register. If you drag the resistors and leds in the breadboard of my drawing to the same places they currently are on your working board that is the first step. Then you need to follow the rats nest lines to connect the leds to the correct pin on the shift register chip. Because the rats nest lines are hard to see I often pick one and click on it and move a bit down it. That causes Fritzing to make a wire. The wire is much easier to see and you can click on it an pull it in to a reasonable configuration. If you don't learn how to do this you probably won't be able to build this on perf board because the wiring is very similar.
Peter