Muellers Hut

After finally getting some time from uni me and my gf are going to do mueller hut in a few weeks and we were wondering A. how early we woudl have to get there to book at the office? and B. if someone could check our itinerary:

Bus into qt, pick up the rental car stay the night with a friend and start driving at around 5am to Mt cook. Park, book the hut, do the hike, dump our stuff and do a couple of the shorter walks nearby. spend the night, hike down, then drive back to qt (her bday) and have a nice dinner. The next day explore (maybe bungee jump) and hit up town in the arvo (but were probably staying near the airport so idk if I go sober or we get an uber), then leave the next day.