Hello, I have used upper wall cabinets as base cabinets. Unfortunately there are no undermount sinks that can be used. If you do find a sink, you will need the p-trap to go sideways. Finally if you are pulling permits this might not pass inspection.

yes, but putting a sink their no.

how wide is it?

is it freestanding? or against a wall?

This is going to be a challenge. I just did a kitchen bar with upper and lower cabs that measured 18" x 72". We decided it was too narrow for a functional sink. With just 12", I don't think it is worthwhile and perhaps not even possible. You should have a plumber confirm and it may depend what is behind that wall (interior or exterior, etc.)