Or you could make sliding tracks for it if you are feeling ambitious and own a router or appropriate somewhat-obscure hardware; but remember to allow for wood swelling/warping with moisture changes in the air.
Making single- or double-hung windows like in a house is a Project and probably not really worthwhile anyhow.
You might consider just trying to find a recycled window. They are left over after people remodel, and are often perfectly good. Many towns have a business that sells recycled building materials, or look for your local "thrift" newspaper, freecycle or craigslist. I've found great windows that were purchased for a home (remodel or new construction) but they changed the design and couldn't use the new windows. good luck.
It is ok to leave the screwed plexiglass exposed on the other side.
If you have a router you could make it so the plexiglas is totally sandwiched between the two pieces of wood, which would eliminate having to drill the plexiglass (which should be done cautiously to avoid cracking), but honestly there is no big reason to have to do that for a chicken coop unless you are just bored