Depending on how they are set up you have a couple of options
Option 1 (Free, Tricky)They are tracking sessions by device, using your MAC address. The MAC (media access control address) is a hardware address attached to each packet used when devices are to be addressed on the same network segment. Let's not get too deep into that, but in a wireless network with no encryption (such as a pay to access network, or in this case your apartment network) you can match the MAC and IP of an already authenticated client and send traffic without issue. The traffic coming from the router will be picked up by both clients but this will not cause any grief as most services are tolerant to this errant traffic. Your Xbox communicates with live by listening on a few particular ports, the traffic destined for the Xbox will be discarded by the laptop as it will not be listening on those same ports, and your laptop traffic will be discarded by the Xbox in the same way as it will only be listening on a few specific ports. Instead of spoofing an existing authenticated MAC/IP to take access, we will spoof the MAC/IP of the Xbox to grant access.
IP Addresses Match- Authenticate @ the web page
- Xbox live should be working
IP Addresses Do Not Match- Disconnect from wireless on Laptop and Xbox (Turn Off)
- Connect Xbox
- Retrieve IP on Xbox
- Manually configure the same IP address for the laptop, then connect to wireless
- Authenticate @ the web page
- Xbox live should be working
Be forewarned if you need to set a manual IP you will probably have to set it back to auto after your gaming session is complete. The apartment router probably wants a DHCP request before authentication.
Now once you have done this once you can leave the spoofed MAC set, and all you have to do is the second set of steps.
Option 2 ($$, Configure once, Depending on hardware they are using may be detectable)- Buy a DD-WRT
- supported router
- Flash DD-WRT
- Set it up in client mode
- Connect laptop to router, authenticate
- Router will route traffic for both Xbox and PC now
Caveat with this method is that the provider may be inspecting for NAT and de-authenticate you.