I'm the proud owner of this guy:
The description is as follows (emphasis mine):
1 bottom drawer, 1 shelf included SHIPPED LOOSE with no shelf clips, the shelf is intended to be field installed to help hold the microwave, this cabinet is intended for built in microwaves not countertop microwaves, must field install microwave, this cabinet does not have finished interior, cut out dimensions are 14 13/16" high by 17 3/4" wide, maximum cut out is 15 13/16" high by 20 7/8" wide, premium UV finished plywood cabinet box construction, 1/2" plywood hanging rails, 5/8" solid maple english dovetail drawer box with BLUM motion integrated easy-close slides
What kind of router and bit is warranted here in order to cleanly and straightly expand the hole?
The only problem with a router would be in the corners they will be rounded not totally square. A very small straight bit would give the smallest Chamfer (non square). This last fathers day my wife bought me this set of router bits I thought oh boy how long are these going to last I usually spend 15-20 per bit. The ones I have used have worked well, I mention this because you could a small bit in the corner then a larger one for the longer cuts. Make sure to use a guide or with a bearing tipped bit have some shims so you get the cut you want.
Straightly has entirely to do with your fence/guide, and there are many ways to do that. You can fence the baseplate, use collars and fence those, or use bearing bits and fence those - it's just a matter of offset from the cut line.
Cleanly is trickier, and depends on where you need clean. Solid wood trim would allow you to cover your sins, while trying to get a perfect cut on both sides of plywood might lead you to a (usually comparatively expensive) double-spiral (upshear and downshear on the same bit) to cut both faces into the middle.
A upshear from the backside or a downshear from the face side is most typical, but may result in tear out on the backside. Unless your guests look at the inside of the cabinet which should be hidden by the microwave being in it seeking tearout, no-one will ever know but you, normally.
I do wonder if the opening is already framed in solid wood strips, looking at the illustration. That would make it much less prone to tearout.
I'm assuming you mean want to expand the rectangular front cut out, and not some other hole. I think most hand-held routers would do the job. There is no need for a plunge feature in this operation, but you will want to guide the router along a straight edge to do a good job.
My first choice would be downward spiral ¼" straight bit, to eject the wood chips from the cut, in combination with a router template guide. This would allow you to follow a straightedge (e.g. straight board) clamped to the outside of the cabinet.
A normal ball-bearing guided straight flute bit could also work. These can be obtained with ball bearing guides on the top (or bottom) of the bit. The ball-bearing guide would allow you to follow a straightedge (e.g. straight board) clamped to the outside (or inside) of the cabinet.
Why are you using a router? Does your microwave overlap the edges? If so, I'd use a jigsaw, seems to me to be much easier set up.