The bad news is we have not done the shuttle yet (soon to do), the good news is my research is positive.
The service is provided by "Commuter Services" and they utilize large passenger van with a baggage trailer. Based on a message left with them and a call back to confirm with them, they will wait/ time the train arrival in Kingman to have someone there at arrival or shortly after, not to say that Kingman at 3 am would not be an interesting place to hang out. Also the feedback is very positive on their drivers and compared to the other option out of LA it is said to work well.
In the travelogue / Trip Reports section "Southwest Chief, April 2014" by Michigan Mom gives a thorough account of this experience.
We are going out (NPV-KNG-LAS) at the end of April (26th) on this trip and due to the late/ early arrival in Las Vegas we booked the time share for Monday into Tuesday. Not knowing how we will feel at that time we would rather have place to lay down even though we are not getting most of that night for the room, a 4 pm check is a long time to hang around, of course it is Vegas so I'm sure you can find a way to win some money or lose your time!