It makes for an interesting thought experiment. It may be difficult in the current version, because I completely hit the sprite limit in the city.
But if I switch to color only mode, I get way more sprites. Also might be possible to do an overlay at the top and use the text system to show a timer or progress bar. Though it'd also be fun to have the palette change over the course of the day, like it's going into the evening.
Might even be able to pause the timer and jump the camera to a hotspot if the player is out of range.
The Game Boy is very harsh with its limitations, but it's surprising how much you can do in it.