I think that sometimes the muscle along the arch of your foot can cramp (this happens to me sometimes, at least I assume that's what causes the sudden pain!). It happens to me really randomly and usually goes away in a minute or less (doesn't seem to be at all related to what I'm doing at the time!).
Arch support definitely helps, and you should try to get shoes with a lot of arch support if you have high arches. I went to a specialty running store, where the owner will watch you run around in each pair of shoes you try on and then recommend the best pair -- the shoes I ended up getting are great, and I'm SO glad I found them! If new shoes aren't an option, I think drugstores maybe sell these brace-like things that go around your foot and ankle . . . I don't know what they're called, but they look kind of like pre-wrapped ace bandages. They will sort of hold up your arch so you don't mess it up too much.
Of course, I'm not an expert by any means
but I do have pretty high arches. Arch support makes a world of difference!