Is there a bus that leaves Harbin and goes at lease close to Changbaishan International Ski Resort?

Yes, you catch a bus from Haxi Bus Station to the mountain. The bus runs every day and leaves around 13:00. BTW, it takes about 7h 30 minutes and CNY160 or so.

Thanks Donna! Is there a way to book in advance given its peak season? And is that a better option than train to Antu then trying to get a bus from there?

As I know, some travel agencies provide the booking service. But they are all in Chinese, so turn to your Chinese friends for help.
As for Q2, yes, it’s better to take a train to Antu first. At Antu Railway Station, there are many private cars can send you to the mountain, and it costs about CNY80 for each.

Thanks again Donna!