Do you mean plug headphones directly into the Airport?
Yes.
Since it's not working using headphones, there are at least four possibilities of what might be wrong:
1) The headphones are bad. One option is to try another set of headphone that you know work.
2) The audio source is bad. If you play the song thru your computer's speakers, do you hear sound?
3) The AX's audio circuity or audio port is bad.
4) The AX is not configured properly.
The following is the basic setup steps for this last item:
AirPort Express Base Station (AX) - iTunes Setup
Either connect to the AX's wireless network or connect directly, using an Ethernet cable, to the Ethernet port of the AX, and then using the AirPort Utility in "Manual Setup" mode, make the following changes:
AirPort > Base Station
o Base Station Name: <rename or leave default>
o Base Station Password: <enter desired password>
o Verify Password: <re-enter desired password>
o Remember this password in my keychain: (optional)
o Set time automatically: (unchecked)
AirPort > Wireless
o Wireless Mode: Join a wireless network
o Network Name: <existing wireless network's name/SSID>
o Wireless Security: <select the encryption type of the existing wireless network>
o Wireless Password: <enter the existing wireless network password>
o Verify Password: <re-enter the existing wireless network password>
Music
o Enable AirTunes (checked)
o iTunes Speaker Name: <enter desired speaker name>
o iTunes Speaker Password: (optional)
o Verify Password: (optional)
o Click Update to write the new settings to the AX
In iTunes:
iTunes > Preferences... > Advanced > General
o Look for remote speakers connected with AirTunes (checked)