There is a shuttle bus that runs from Nairobi to Arusha - twice a day - as well as other buses. You won't have a problem organising the transfer when you arrive.
If you book your kilimanjaro climb with an operator in Arusha before you arrive they can organise and book the transport from Nairobi for you.
Our Travel Center has a good choice of itineraries for all routes up Kilimanjaro.
Have a browse through - click links for detailed itinerary...
http://www.africaguide.com/rd/climbing.htm