Hi Tony,
Based on my research, Exchange supports moving your public folders to Exchange Online from Exchange Server 2007 SP3 RU10 or later version. For detailed information, please refer to the article below. In addition, you also can export the data and import them to the public folders manually.
Migrate Legacy Public Folders to Exchange Online
http://technet.microsoft.com/en-us/library/jj983799(v=exchg.150).aspx
If a public folder mailbox reaches the Warning quota of 24.5GB, an automatic public folder move will be triggered. Some public folders will be moved from the nearly-filled public folder mailbox to another public folder mailbox with enough space. If there are no public folder mailboxes with enough space, Exchange Online will automatically create a new public folder mailbox and move some public folders to the newly created public folder mailbox.
For detailed information, please see the article below:
Public Folders and Exchange Online
http://blogs.technet.com/b/exchange/archive/2013/05/02/public-folders-and-exchange-online.aspx
If you need further assitance, feel free to let me know.
Thanks,
Aaron Yuan