cblaisd, welcome to Los Angeles, albeit for transitional purposes only.
Connecting to your Sunday morning flight should not be a problem. I'm concerned about your return connection on New Year's Eve. Your return UA flight connection could be tight, however with an on-time LAX arrival on HA and not having to claim luggage and then re-check it, it should be possible (with a little luck). You didn't mention your luggage situation. Hopefully, the airlines are transferring it?
I would suggest walking (Ground Level) between terminals rather than waiting for the Terminal to Terminal buses. Curb to curb it's about a 5-8 minute brisk walk between UA (Terminals 7/8) and HA (Terminal 2). With the wait and the frequent stops, I believe the shuttle bus would take considerably longer. There are sidewalks all the way, along with a couple traffic signals. You'll pass near Encounters Restaurant as you take the shortest route between those two terminals.
LAX Terminal Map
http://www.lawa.org/lax/terminalMap.cfm
Aloha, happy travels and best wishes with your connections.
