If all you want to do is see Kilimanjaro in the distance you can just drive out along the A23 a bit. Easily doable in the time you have. That's as close as I got so I'll let someone else add a more detailed answer.
The closest point of the A23 to Kilimanjaro is about 80km from downtown Arusha, and the road is a decent one. My memory is hazy, and we weren't coming from Arusha, but 1-2 hours would certainly get you there. Remember that is just to see Kilimanjaro in the distance.
If you are going to do that, it might be worth visiting Kilimanjaro National Park, at 128km from Arusha. I've never visited but there are day hikes available.