There's 3 ways to do this, from what I can think of:
1) Forward All Traffic. This would effectively bounce all traffic from your phone to a server, then server to your browser. This would skyrocket the cost to the AirDroid team since they'd have to buy a server with quite a bit of bandwidth. Unless you guys are willing to pay for it, or accept ads (I'm not -- any permission granted to the app could also be available to the advertising source and I'm not comfortable with that). This option presents the least headache to the user (it'll be "web.airdroid.com/user" or "user.airdroid.com"). This option would also have to be implemented by the AirDroid team, or at least require a user to be able to configure routing paths on their router (i.e. unlikely / not user friendly)
2) Port Forwarding. You should actually be able to do this right now. If you go into your router and open a port directly to AirDroid then configure a DynDNS or no-ip dynamic IP address, everything should be good to go. AirDroid, according to the description, already has a single - login / rate limited password that transmits over HTTPS. However, this option REQUIRES you to be able to configure the router. i.e. probably just for home.
3) VPN. You create a VPN connection between your computer and your phone. Either you'll install a VPN server on the computer you're using, or a home computer that both your phone and the computer you're browsing from. This requires more technical knowledge than #2, but should be able to accomplish this with more reliability as it doesn't require router access -- just being able to configure a VPN client on the browser computer. You could set up the VPN host on your phone, but I think that might require rooting. A quick search returns mostly clients to paid/free services. Those with Windows Professional (any version) will be able to configure a VPN server for no extra cost.