Yes it's possible. I'm using a similar setup now. I live in a large house that was gutted and turned into separate apartments. Our landlord has a router in B with cables from it going to units A and C where I live. Problem is he forgot the wifi password. After trying everything I could think of, I just brought out a spare router I had, and plugged it in. I had to log into it and turn it into a wifi access point. Units A and B are the PC in your pic, while my laptop/phone is wifi over other router, and my PC is plugged into the back of it. Very doable.
Slow down? Maybe a little, but nothing you should notice. My ping in games is still good.