Well, you actually don't HAVE to set DHCP manually - it will probably default to 10.0.1.x. But what you want to do is make sure you're not using the same subset that the hotel gives you.
To check, connect JUST your Mac to the hotel ethernet, open Network preferences, and check your IP address. If it's 192.168.1.x, then use the 10.0.1.x range as DHCP on your Airport. If it's something else, just make sure you're using a different range on the Airport.
Connect the Airport, run Airport Utility, and make sure that under TCP/IP, it itself is getting a sensible IP address from the hotel (usually setting ITS connection to DHCP is sufficient).
Then configure its own DHCP tab appropriately. Begin at x.x.x.2 (because the Airport will be x.x.x.1) and end the range at x.x.x.10. Apply the changes.
Let me know if that works!
Matt