how to solder MEMS microphone to PCB at home?
One option would be to buy a hot-air reflow station, e.g. Sparkfun. Chinese-manufactured ones are reasonably inexpensive, and some are of good enough quality for prototype/home use.

You might also consider dead-bug prototyping with some 30-gauge wire or so and a good quality hand solder with a fine tip. This involves turning chips upside down and soldering pins directly to parts or using wire, without necessarily using a board (you can solder the wires to a PCB or a copper-clad board of some kind if you want). This can require fine wire and a lot of skill for parts with small, fine-pitched pins.