Sounds like you don't regard UA as having a 1st class, ha ha, but it's a chance for award tickets. If you pay double the miles ("Standard") you are allowed to pick any day, otherwise the normal award (called "Saver") you have to pick a day with availability (several months out, usually, or 1-2 days ahead).

UA's website lets you check award availability, best to register yourself as an "expert" and see all availability.

I fly IAH-SYD four times a year, have found the optimum is to fly Continental to LAX and transfer to UA to SYD. Of course UA's award would include the IAH-DEN-LAX sectors at no extra cost, just be very careful with the connections as if you misconnect you are not going to get F seats on the next day's flight to SYD. Would recommend departing IAH 4pm on UA or 6pm on CO. Coming back through DEN is horrible, always delayed, best use CO.