I can work in a two-day layover in either Rome or Madrid en route to Cairo (and one day on the return).

I'll be pretty jet-lagged (24 hour flight!), and trying to keep things on a serious budget. It'll be: wander the monuments in a jet-lagged daze, then maybe hit a club or sauna.

Which would you guys recommend?

Having been to both of those cities, I would choose Rome anytime.