As regards what you intend to do, it will work and may give you better wireless coverage.
Good wireless signals depend on many factors such as the construction of your property. The thickness and construction of the internal walls and if they have any form of insulation within them which can impeded wireless signals so it is not possible to say if your wireless signal will definitely improve or not.
Based on the excellent performance of my BT Complete WiFi in a much larger house than yours I would say think again. A single black disk placed somewhere near the head of the stairs should deliver a good signal to point B and, in all likelihood, to the rest of the house too.