How can I remove a mouse stuck in the air duct above a kitchen exhaust fan with built-in microwave?
Where there's a mouse there are mouse droppings and urine... I'd get that microwave out and clean it, if not replace it.

It looks like there are a pair of screws in the front corners of the cabinet going down into the microwave. If so.. those are probably what's holding up the front edge of the microwave. The rear edge is probably supported by some wall-mount bracket.

Don't forget to unplug the power cord. Then remove the two screws at the front and allow the unit to tip downward; after tipping it downward a bit you'll be able to lift the rear off of the wall bracket. It can sometimes be managed by one person, particularly if one knows how the assembly works.. but that not being the case here, recruit help to deal with the weight and any unanticipated fasteners/challenges.

Make sure both you and the assistant are comfortable with the idea of a spooked mouse dashing out of its prison, over you, and disappearing into the kitchen.