If you only plan to share the database with one other user you don't really need Filemaker Server and can share it directly from your Filemaker program.
If you do have Filemaker Server 5, you can use this as well.
If you plan to buy Filemaker Server now, you will need to upgrade both your Filemaker clients to version 9 as well. The database will also need to be converted using Filemaker 9.
Sharing the file over the Internet is pretty straightforward, but can be challenging to set up in certain situations.
You need to enable sharing for the file in Filemaker Pro.
Your partner needs a copy of Filemaker Pro. (version 5, 5.5 and 6 are compatible with your database without conversion). If he needs to buy a new copy of Filemaker (which is version 9), then you also need to get version 9 as 9 is not networkable with 5.
TCP port numer 5003 needs to be open on your local internet firewall/router and forwarded to the machine hosting the database, which should have a static ip address.
Your partner then needs to enter your internet IP address in his Open remote dialog in Filemaker to be able to connect to your machine.
Make sure you use passwords in the database before you open up for outside access.
Lastly, I will mention that you may be able to use Instant Web Publishing (IWP) to let your partner have limited access to the file through a browser, bypassing the need for Filemaker at his end.
You may want to have a look at this resource if you are considering IWP for Filemaker 5:
If you don't want to keep your machine running all the time, have to deal with static IP addresses, etc. you may also want to consider a FileMaker hosting service like ours: