Can I increase the efficiency of a refrigerator by replacing the door insulation?
Yes it would.  Extruded (tiny bubbles, no beads) polystyrene runs about R5 per inch, compared to R3.5 for fiberglass.  

However:  The door is only about 1/5 of the surface of the fridge.  You are going to make that 1/5 possibly have 1/3 less heat loss.  So overall you are reducing it by 1/15 or about 7%  

In addition transfer thorugh the walls is only part of the heat loss.  More is caused by opening and closing the door.  If half is caused by loss of cold air on opening the door, then your savings just dropped to 3.5 %

If it is old enough to require rust repair, it may be time for a new fridge.  Fridge is one of the big power users in the house, and cooling systems have improved a lot in the last 20 years.