You might need to replace the top half of that cupboard with a unit that's designed to be partially a microwave and partially cupboard -- otherwise, I guess you have to find something that fits within the depth of the cupboard to let the doors close, which seems many kinds of inconvenient, or just to take the doors off and have open shelves above.

You don't want an over-the-range microwave, but a countertop style -- then you can either just have the shelf size a little larger than the microwave all the way around, or you can get a kit to frame out the face and make it look "built in" as you prefer. (compare the 2nd and 3rd photos before)

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There are some units specifically designed to be built in, but they're likely to be more demanding than you want with your available space/cabinet. Look for one that will work with the depth and overall size of the cabinet you have, so you only need one kind of retrofit (the doors). Also don't forget that you need to have an outlet built into the cabinet rear too.

We just replace our OTR microwave with a ss hood. We bought a small counter top microwave and put it in one of our lower double door cabinets. Works great, actually better than having to reach over the stove.

I would definitely recommend to get a microwave over your stove area, its so practical and it acts as a hood as well. I have it like that in my house, and it has worked out great so far. Also the microwave has lights underneath, that at night you can leave on and it gives it a nice "night light" for the kitchen area if you go up during the middle of the night to get water or something.

There are quite a few different microwave options! Here
is a great article: Where
will the Microwave go?

You should not put a microwave in a cabinet ... it needs to be vented. From what I can see your choices are to bring in a carpenter to adjust the cabinets, use a counter top one, or create another space for it on something freestanding.