Unfortunately, no. Both wireless and wired traffic coming/going to the Internet will go through the AirPort's NAT firewall.
However there are some configuration options you may want to explore to see if they can help.
Basically what it comes down to is that although you can get an Internet connection for the Xbox with the AirPorts, you may not get the necessary NAT setting (Moderate or Open) for the Xbox Live! on-line game that you want to play. As such, you basically have two options: 1) Create a DMZ (Apple calls this a Default Host) for the XBox or 2) Configure the AirPort for Port mapping to open the required ports to allow Xbox Live! access ... or 3) Use a compatible router.
Option 1 - Create a DMZ
(Note: This option completely exposes your Xbox console to the Internet. You may want to consider option 2 instead if you are concerned about the risks of doing so.)
AirPort Utility > Select the AirPort > Edit > Network Options
Enable default host at: <Enter the IP Address for the XBox>
Option 2 - Configure Port Mapping
The following ports must be available for Xbox Live to operate correctly:
UDP 53
UDP 88
UDP 1863
UDP 3074
TCP 53
TCP 80
TCP 1863
TCP 3074
To setup port mapping on an AirPort base station, either connect to the AirPort's wireless network or temporarily connect directly, using an Ethernet cable, to one of the base station's LAN ports, and then use the AirPort Utility, to make these settings:
1. Reserve a DHCP-provided IP address for the Xbox 360.
AirPort Utility > Select the AirPort > Edit > Network tab
Click the "+" (Add) button to enter a DHCP Reservation.
Description: <enter the desired description of the host device>
Reserve address by: MAC Address
MAC Address: <Enter the MAC hardware address of the Xbox or the MAC address of the wireless depending on how you connect the XBox to the network.>
IPv4 Address: <enter the desired Private IP address you want to assign to the Xbox>
Click Save.
2. Setup Port Mapping on the Express.
While still on the Network tab:
Click the "+" (Add) button under the Port Settings dialog box.
Description: <you must enter a description here; it can be anything you want>
Public UDP Port(s): 53, 88, 1863, 3074
Public TCP Port(s): 53, 80, 1863, 3074
Private IP Address: <enter the DHCP Reserved IP address for the Xbox you created earlier>
Private UDP Port(s): 53, 88, 1863, 3074
Private TCP Port(s): 53, 80, 1863, 3074
Click Save
Click Update and allow the AirPort to restart.