You didn't identify your hardware, so the first step would be to find out if it's supported by the latest OS.
Yes, in order to gain access to the App Store, you'd need to start by getting the system upgraded to 10.6.8 (Snow Leopard.) From there you could purchase Mountain Lion and OS X Server (It's a standalone application now.)
Additionally, I don't believe there is a direct upgrade path from 10.4 to 10.8. So I think you'd need to get 10.6.8 Server, perform the service migration, and then go to 10.8 and perform the migration again.
>> It's a 2 x 2.66 GHz Dual-Core Intel Xeon Mac Pro.
I'm pretty sure that makes it a MacPro1,1. Even if it is a MacPro2,1 it still won't run Mountain Lion. The best you're going to do with that system is Lion (10.7.5).
If you can get the OS on a thumb drive, that would allow you to skip the 10.6.8 process... especially if you don't care about migrating your data.
Unless you're using the extra drive bays, or have something like Fiber Channel cards installed, a simpler solution might just be to buy a new Mac mini server.