What's the range on a typical $20 USB Wireless adapter

That's going to be close and a lot to ask of a router and adapter but if you have a dual band router and dual band adapter then you have two frequency bands to try and you have several channels also so with all those options you may be able to get signal. It will also depend a lot on the antenna of the adapter and the anatenna of the router also.
If the signal is weak or you can't connect you can try a range extender to get the signal on to the second floor.