While navigating to Edit screen from Settings screen, use this (inside your Settings screen)
Navigator.pushNamed(context, "/editScreen").then((_) {
  // you have come back to your Settings screen 
  yourSharedPreferenceCode();
});
If you only want to run this code on some event that happened on your Edit screen, you can make use of pop method inside Edit screen like
Navigator.pop(context, true); // pass true value
And above code should be:
Navigator.pushNamed(context, "/editScreen").then((value) {
  if (value) // if true and you have come back to your Settings screen 
    yourSharedPreferenceCode();
});
Edit:
async-await would look something like:
bool value = await Navigator.pushNamed(context, "/editScreen");
if (value) // if true and you have come back to your Settings screen 
  yourSharedPreferenceCode();