FPS character, full body or just arms?

I'm pretty sure it would -technically- go faster as just the arms. How much faster? Nothing you would notice unless you're animating a complex body (will still have to do calculations even if you can't see it). Even then Unity and modern GPUs are pretty powerful, you would be hard-pressed to make a negative impact. If in doubt just try both and see what happens.

I would always prefer using just the arms though it completely depends on the complexity of your character and the platform you are targeting. If you want to deploy on IOS or Android, please use the arms only without doubt. Also Occlusion Culling wouldn't really be of much use here because it only works with whole objects not parts of meshes :)