If you only need to use this service from inside your office network, you don't need to do anything to the AirPort Express at all. From your server machine, go to "System Preferences > Network > Ethernet" (or the AirPort interface, depending on how it's connected to your network), and see what IP address it has. You should be able to connect to that address from your client software on any of the clients on your office network.
If you need people to be able to connect to it from outside of your office network, it gets more complicated.
People inside the network can still use the private internal IP address of your server that you just looked up. But for people outside the network, assuming your Express is doing NAT (we'll get to that in a moment) you'll need to set up that port mapping and have them connect to the public IP address of your Express, which you can find via the AirPort Utility.
The AirPort Utility is in your /Applications/Utilities folder. After you launch it, your AirPort Express should appear in the list on the left. Double-click it (or choose "Manual Setup") to view and modify the configuration of your Express.
Now go to Internet > Internet Connection and look at the "Connection Sharing:" pop-up. If it's set to "Share a public IP address", then your Express is doing NAT, which is what I would expect. If it's set to anything else, then you've got a potentially simpler office network setup, and the rest of these instructions don't apply.
Now, under "Internet > TCP/IP", you can find the public IP address of your Express. This is the IP address that people outside your network should point their client software at, so write it down. Hopefully it has a static IP address that you've configured manually. If it's set for DHCP, it may mean that your ISP can change your public IP address at any time, which makes it much more complicated for you to provide a service that people can reach from outside of your office network. You might need to find out if you get any static IP addresses with your business-class broadband Internet service, and if not, see what it takes to get one. You might end up paying a little more per month.
Aside: AirPort Expresses also support "hairpin NAT", so even from machines inside your office network, you should be able to point your client software at this public IP address and end up connecting to your server. However, this causes their traffic to cross your network twice and could affect your Express's performance, so it's more optimal to point your internal client machines at the internal IP address of the server.
Now go to "Internet > DHCP" and see what range of IP addresses your Express is serving. By default, an Express will use 10.0.1.1 for itself, serve out 10.0.1.2 - 10.0.1.200 via DHCP, and leave 10.0.1.201 - 10.0.1.254 for static, manually-configured IP addresses for things like servers. If you're using a different IP address range here, then you'll have substitute addresses appropriate for your custom setup into the instructions below.
Now go to "Internet > NAT" and either set 10.0.1.201 as your Default Host (Apple's term for a DMZ) or go to the "Configure Port Mappings" to specify just that one range that your server software package uses. I'm picking 10.0.1.201 as the private static IP address you'll be using for your server. If you've already picked a different .201-.254 IP address for your server, substitute that address in these instructions.
If you made any changes to your Express's config, don't forget to hit the "Update" button to save them to be Express. While I'm on the subject, don't forget to update your Express's firmware while you're here. A modern 802.11n-capable Express should be on firmware v7.4.2. And older b/g Express should be on firmware v6.3.
Now go to your 10.5 box that you're using as your server and manually configure its IP address settings in System Preferences > Network > Ethernet (or AirPort, depending on how you have it connect to the network) > Advanced > TCP/IP. Give it IP address 10.0.1.201, subnet mask 255.255.255.0, default gateway/router 10.0.1.1, DNS server 10.0.1.1 (this tells it to use the DNS proxy in the Express, or you can give it the public DNS server addresses from your ISP, or OpenDNS addresses, or Google DNS addresses like 188.8.131.52 and 184.108.40.206). For search domain, you probably want to use your business's domain name. If you made any changes, hit "Apply".
Now make sure you've got your new server software running on your server.
That's it, you should be done with the setup. Now you can test it. From a client machine inside your network, run your client software and try connecting to the server via its internal 10.0.1.201 address. If that works, then from the same internal client, try connecting via the Express's public IP address. If that doesn't work, it could just be a bug in the Express's "hairpin NAT" code. Try having someone outside your office network run the client software and try connecting to your Express's public IP address. That might still work even if the hairpin case didn't.