I'm sure you have Presta inner tubes, these are really more common on lightweight road bikes (and Mtn. bikes) than the Schraeder valves you are more familiar with (like on car tires). You just need to get a Presta compatible pump, and ALL the support staff on the Aids Ride will have pumps that fit these tubes, have no fear.
The '93 Allez was a pretty nice bike, Taiwanese and pretty "tight" geometry, but a well-built bike and a good value.The RX100 stuff might be showing its age and may need refreshing, but maybe not if it's really been stored for years...otherwise if it fits and you find it comfortable for 8+ hours in the saddle, then you did fine with your buy. I'd suggest trying to fit the fattest tires you can get away with in this frame for increased comfort on the road, but it may have such tight clearances that you can't even get a 28mm in there without rub.