Long answer: 

You provided an awful lot of detail about the machine, much of which is irrelevant to whether it can run a Processing sketch (it does not matter that it has FireWire, a headphone jack, etc...).  You did not provide one highly relevant bit of information: What's the sketch?  Some sketches will perform better than others, and the only way to know for sure how well any particular sketch will perform is to TRY IT.  If there are

performance problems, it is generally a better idea to fix the sketch than to attempt to fix the machine.

One issue that can lead to problems are 3D sketches that require OpenGL running on machines with poorly implemented OpenGL drivers or poor performing OpenGL hardware.  This occurs more frequently on low-end PC hardware with integrated graphics hardware..  Even first gen Mac Minis, however, have pretty decent OpenGL support, so that shouldn't be a problem.

I don't see any major red flags in the copious specs you provided.

Thanks again for your response.