Remember Gaaras automatic defence when he was still a Jinchuuriki?
Well it's almost the same with Obito and Kamui.
His intangibility works like this, Obito sees or senses an incoming attack, during the incoming attack his whole body is in a state of caution, it still exists in the physical realm and it still has physical properties, however during contact the affected body part loses it's physical properties and is send away to the Kamui dimension.
Sure he does have the sharingan and he's able to predict movements to some extent, but he still needs sometime to react and even if it's just a brief second, that's just enough of an opening for some people in the NV.
Where as Kamui or his intangibility, to be more specific, is instant.