Making Airport appear as a computer I live in a building where I am connected to the university network for internet. I am not allowed to connect a router, or I will be kicked off the network. (This rule was created after wrongly configured routers blocked internet access for the entire building.) They expect people to connect their computers all through ethernet.

I have multiple computers including a MBA and an iPad, so I don't plan to abide by this.

I know I could configure one of my Macs for internet sharing, but it would have to be running all the time. Since I own an Airport Express I was hoping I could use it instead. Is there a way to configure the Airport so that the network administrators can't detect I'm using a router?
Have you tried talking to your campus' Network Administrators? Plead your case! If they ask why, hold up your iPad. You are never going to be running a cable to that thing with the intention of using it on a network. Only power and maybe headphones!

Working with them might let you use your wireless with their recommended settings. Maybe bring the Express with you and let them configure it in the manner they want. Maybe they would prefer you NAT yourself, maybe they would prefer you to Bridge the connections.