This computer is more than enough to run a dedicated server for minecraft.
You should check the minecraft server wiki. Running a server is different from running the actual game. A server shares the data but doesn't process graphics, so it doesn't need to be very powerful and the less heat it generates the longer it will last. I would build the system I recommend, the server wiki say's that it would be strong enough to host 45+ users. The money you save would go toward getting a huge hard drive if you want as well.
Like this:
http://www.newegg.com/Product/Product.aspx?Item=N82E16822148844
that is 3tb and would still make the system cost less than the $400 budget even with the 64gb ssd running the OS and MC files.
This is the minecraft server wiki.
http://www.minecraftwiki.net/wiki/Server/Requirements/Dedicated
This is the optimal server, which means what they think is the best when running Linux.
Optimal
45+ players: Intel Core-Based CPUs or AMD K8-Based CPUs and better, 6 GB, 35 GB empty space, 30 Mbps down, 15 Mbps up, A SSD is preferably recommended for a populated server.
If you are really concerned about performance I would get the 2.13ghz Atom processor. But I honestly don't think you would need all that power.
http://www.newegg.com/Product/Product.aspx?Item=N82E16813500074
You would want to run a dedicated server, which means all it does is serve the game 24/7 if you want. You would use another computer to connect to the server and play the game. All this server is doing is sharing the files needed to display the map with the users that are connected to the server. So running the server is not as "demanding" as you might think. For example, I can run a server for a very demanding game like ARMA2 on an intel atom set-up because it is not running the actual game, but just transferring information about the game to computers that are actually running the game. ARMA 2 won't run on Linux but I can run the server on Linux. The dedicated server recommendations would never run the game even on the lowest settings but it will run the server with over 32 players.
I hope that example helps.