Are the two flats on the same electrical service? You might be able to arrange Ethernet-over-powerlines (e.g., http://thewirecutter.com/reviews/best-powerline-networking-kit/) for getting data back and forth, and at that point, you just need the right gizmo.

Searching Google for "Ethernet doorbell" actually turns up a number of hits. You can kludge something together from Arduino parts (http://lifehacker.com/5908850/build-an-a...-text-and-photo) or maybe buy something that suits your needs. There seem to be a lot of "video doorbell" contraptions out there, which might be the proper "overkill" solution to your problem.