For a 20" rear wheel with a this kind of fender, what would be the best angle to install it to minimize the spray on one's butt/legs?
The spray will come off at a tangent to the wheel. If you're trying to prevent your back getting wet, putting the short fender to the rearmost position is the best but your legs will still get spray. If you want to prevent your butt getting wet, raise it a little. Note that going faster will mean the wind will push the spray back a little, but the spray will also increase in velocity and strength.
Unfortunately, your fender on your bike as-is is too short to protect your entire body and legs. You could make your own fender extension using milk jugs, wood, metal sheeting, plastic sheeting, coroplast, or any other number of DIY materials.
Note: Image is from a website selling a fender for bicycles.