If I were to purchase a VPS account, will I be able to run a small game server program off from it, custom port and all? I am asking because I am currently writting an Client/Server Game Application, and I need a server to host the Server software, and the server cannot be behind a router. Thanks. 
Should be no problem, the only reserved port that I know of is the SSH one (port 22). You could reconfigure even that I guess, but I wouldn’t recommend it 
I run a couple of applications on custom ports without problem.