First thing please be aware a Microwave over the stove generally draws 10-12 amps. The fan hood you removed had a 1 or 2 amp draw most likely. Fan hoods are almost never on a dedicated circuit so connecting it to that line you may find the circuit overloading and tripping the breaker. Consider running a new circuit if possible.

As for how to terminate the wire. You need to install the receptacle inside the cabinet above the Microwave. You can use a surface mounted 1900 box with a mulberry cover for a GFI which by the way this should be.

Hope that helps

I hate to state the obvious, install a receptacle in the cabinet above. Electrical equipment & appliances are required to be installed in compliance with the National Electrical Code and the instructions that come with the appliance.

Is this a combination microwave & exhaust hood? Or are you removing the hood & installing the microwave? I'm trying to understand why the exhaust hood wiring is getting involved in this project.

Yes it's a combination hood/microwave, and yes I'll be removing the old. I'm going to test if it's on its own circuit the fan that is, if not I'm sure it's on a line with one or some of the receptacles on my countertop. If that’s the case I'll probably just avoid using those receptacles instead of having a whole new line run. Would that be ok? Or will I still have a circuit problem?

If you know what's on the circuit and avoid overloads it should be okay. You will not be able to run the micro at the same time as some other appliances if they were on that circuit like a dish washer, toaster, coffee pot etc.

If it turns oput that the existing circuit serves the kitchen receptacles, then you will need to run a new circuit . If the circuit does not serve the kitchen receptacles and the load is more than 50% of the existing circuits capacity, you will need to run a new circuit.

Tom