There is a game that is coming out soon is called Phantasy Star Online 2: New Genesis, and I am unable to play it.

Is there a processor upgrade that I can install that has the AVX instruction set? The current processor does not have AVX.

Unfortunately no.  I don't think AVX support started until Intel launched the Sandy Bridge CPU.  Those CPUs were supported on the S/C/D30 series ThinkStation platforms.  S20 ThinkStations only supported Nehalem and Westmere based Xeon CPUs.  If you need AVX2, I think you'd need to be on a Haswell (or later) Xeon CPU.  Those weren't supported in ThinkStation until the P500/700/900 series platforms.