I'm not a machinist but I have taken micrometers apart. Merely unscrew the top part. The your PB Blaster or whatever can reach inside where the threads are. Sometimes they can be a bit finicky to get back together. Turn the threads backwards until they click in place then you can turn them forwards to assemble.
Micrometers are made to be taken apart for cleaning and lubing. The should have very light oil on the threads. Starrett actually makes it's own micrometer oil. I use it for anything that needs a light oil.
I use micrometers for rebuilding transmissions in my hobby motorcycle work.