I plan go on this december start from beijing to jilin - changbaishan-harbin or is better how?

Your route is okay, and I’d like to give you some suggestion.

Take a bullet train from Beijing to Jilin, two trains from Beijing Station leave at 6:39 and 14:28, the other two from Beijing South leave at 7:30 and 17:00. It takes about 6-8 hours, CNY285.5/469.5 for a second class seat, CNY375.5/680.5 for a first one.
After that, you can hire a local car to the mountain, which may cost CNY800-900. The fare can be shared with other visitors by the way.
Then to Harbin, take a local car to Changbaishan Airport, which costs around CNY200. And then take the flight at 16:35 to Harbin. It takes less than 1h.