I currently have an OTR fan/light combo that is on a dedicated 15-amp circuit. I would like to install an OTR microwave that requires a dedicated 20 amp circuit. Is it appropriate for me to replace the 15 amp circuit breaker with a 20 amp, and run new 12 gauge dedicated wiring from the circuit breaker to the new microwave? Should I use armored cable since I cannot ensure proper routing of the wire through the wall?

The way you suggest is the way to do it. You can use NM (Romex), AC, or MC cable for this. None of these have to be supported if fished through concealed spaces in finished buildings. Keep in mind that they must be supported where you are able to do so without ripping walls down.