Yeah, I made a thread earlier on: Would Superman be more capable than Batman in fixing Gotham's problems?
Now I was thinking about a reverse scenario. What if Superman disappears which is something like the beginning of Superman Returns (2006) movie, and Batman decides to step in to cover for Superman - Assuming that Batman had left already Gotham in the safe hands of another protector (So Batman need not worry about both Gotham and Metropolis at the same time).
How would Batman's prep, abilities, character and personality fare in solving the problems that Superman regularly faced in Metropolis? If Batman was in Superman's place, do you think he would perform better as the protector of Metropolis - than Superman ever does?