The Thredbo YHA told me that it is possible to get there with a bike using the school buses. The staff at the YHA said that others have done this with bikes without any issues.
They provided the following info/timetable:
BUS Canberra to Cooma
$16.66 pp one way
Countrylink
Depart Civic, West Row 11.55am
Arrive Cooma 1.30pm
(Mon, Wed & Fri)
Depart 4.20pm - Arrive 6.18pm
(Every Other Day)
Return:
Depart Cooma 10:05am
Arrive Canberra 12:03pm
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BUS Cooma to Jindabyne
$20 pp one way
*Cooma Coaches 64524841
Mon to Friday – not during school holidays
Depart Cooma 2:15pm
Arrive Jindabyne 3:20pm
Return:
Depart Jindabyne 8.55am
Arrive Cooma 9:45am
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BUS Jindabyne to Thredbo
$20pp one way
*Snowlink - School BUS
Mon to Friday – not during school holidays
Depart Jindabyne 3.30pm
Arrive Thredbo 4.20pm
Return:
Depart Thredbo 8:10am
Arrive Jindabyne 8.55am
So it really looks like it would take a whole day to get there along with multiple transfers with your bike.
You also might want to confirm with Snowlink and Cooma Coaches who run the school bus services if you can take a bike along. You may also need to ask if the bike needs to be boxed, etc.
Also, obviously the above bus services do not run outside the school holidays. Currently Jetstar have $130 flights to Auckland. In the time it would take you to get from Sydney to Thredbo via the above, you could fly to Auckland and then get a bus to Rotorua. It might even work out cheaper as well.