I am both. I'm not so much obsessed with time in a way that makes me care whether a watch is accurate down to the exact second, but more so in a manner dealing with punctuality. I can't stand being late. Especially in business (whatever your business may be), I think it is vitally important to give the impression that you are prepared and ready to go, right on time in the right place, and that handling the task at hand is your highest priority. When I am late I get a nagging feeling of unpreparedness, and a feeling of being out of control. I usually arrive places about 15 mins early, and then wait outside if its somewhere that being early would be unfashionable. I am like this in other settings as well. In college, I would always make sure I was early to class, so I could get a choice seat (in the back). I never wanted to be that guy arriving late, flustered, and fumbling around trying to find an empty seat while everyone was looking at him like he was interrupting the beginning of class. Even at the movies I like to get there early and have my pick of seats, rather than sneaking in, in the dark. Idunno, maybe I'm a bit of a control freak. ;-)
I'm also fascinated with the concept of time, especially recently with the theoretical idea that all moments in time (past, present, and future) exist permanently just like every location in space exists. That our concept of the present time is just an illusion due to our limited understanding from our human perspective. I'm not sure if I believe this to be the case, but I am amazed by the idea. The idea that when we cause or experience joy, those are moments that are existing forever. Or transversely, when we hurt the ones we love we are hurting them in that moment throughout eternity. I'm no Einstein, so I'm sure my understanding of such theories is flawed, but I think the concept is such a beautiful idea.