Ok, so if I understand you correctly, there is no "relay" configuration possible.
Correct, in the "extend a wireless network" feature.
My computer guy originally set it up as a "slingshot" system designed to relay a signal upstairs. Now it seems that that is not possible?
This sounds like he used an older WDS type of configuration, which would allow a "main", "relay" and "remote" type of arrangement. There are two huge downsides to WDS:
1) The system will only work at "g" wireless levels, so you lose any of the faster "n" capability of newer routers and devices on the network. You new "n" AirPort Extreme and AirPort Express instantly become older "g" devices with WDS
2) The bandwidth penalty of WDS is enormous. Each relay or remote on the network cuts the available bandwidth in half. So the relay cuts the speed of the "g" wireless network in half, and the "remote" cuts that half in half again. After you do the math, you have a "g" wireless network that is running at 25% speed.
I think it's pretty clear why you would not want to even consider this type of arrangement.
If not, then I should use the new Express in the foyer rather than my old one and hope that the signal is strong enough to make it upstairs.
The "ideal" location for an extending Express (assuming no major obstructions in the signal path) is a location that is about half the distance from the main router to the area that needs more coverage. Remember that the Express can only extend what it receives, so if it is located too far from the main router, it extends a very weak signal. You will need to experiment a bit to find the best location in your particular situation.