You have to arrive into GUM when Continental Micronesia flies to Cairns. If I'm correct, Continental Micronesia flies to CNS on Sunday and Thursday while they leave CNS to GUM Monday and Friday. So for example, if I got the frequencies to CNS correct, flying from LAX, you'd have to leave LAX on Saturday, May 24th, crossing the dateline into GUM on Sunday, May 25th. Then leaving CNS on Monday, June 2 which will then take you through to LAX. So in summation, you have to depart when the frequencies of the GUM to CNS to GUM flies operate. Hope I didn't confuse you.

BTW, your routing will be LAX-HNL-GUM-CNS then return CNS-GUM-HNL-LAX.