Utility Sink and Washing Machine Drain Pipe
Yes, the picture you included shows a good plumbing layout for a washer and laundry sink. In particular, it provides good ventilation for both the washer standpipe and laundry sink, and accommodates a P-trap in both locations (only the washer trap is shown because the sink trap will be installed outside the wall). Another advantage of this layout is that it uses a 2" PVC line for the washer drain, which is code for washers because a more powerful washer drain can overwhelm a 1 1/2" pipe.

You are working under some tighter constraints but it looks like you should be able to implement a similar layout. You might want to draw it out and see if you can achieve workable slopes (min 1/4" per foot) and get the vent pipe to the right height (above the overflow height of the sink). It may be helpful to swap the washer and dryer locations -- you should also switch the flexible dryer vent out in favor of some rigid ductwork.