It does look like a Mock Orange. You can prune now unless you want flowers. Prune after it blooms.
You can cut them as low as you want to attain the size/shape you want. But, you may sacrifice next year's flowers as they bloom on last year's growth.
I think most people maintain by pruning (after they bloom) according to the "1/3 Pruning" rule but that is way too much to think about. So I leave mine alone until it gets silly and then cut the whole thing to the ground.