If the hotel is giving you a private IP in the same range that the aiport is using for DHCP then you could have problems.
Try changing the Airport's settings to assing address in a different range.
For instance, if you are using 192.168... then try either of these:
172.15.1.x or 10.1.y.z

While its unlikely, they can actually detect the airport express and disable the ethernet connection when you try and use it.