With Exchange 2013, and so Exchange Online too, a new feature has been introduced with OWA (Outlook Web Access) called Offline Access – if you don’t how to use it, take a look at my post here http://blog.hametbenoit.info/Lists/Posts/Post.aspx?ID=393
That said, this feature has been improved as now you can select WHICH folders to synchronize when you are using the offline access for OWA. This is great as until then (and this is still the default setting), only inbox, draft and calendar where synched.
To choose which folders (in addition of the default ones mentioned), you must (off course) have enabled to offline access.
Then, just hit the Gear on top right of the window (on the right side of your name) and click on Offline settings
Then click on the offline folders link shown and enable the selection by hitting the Choose folders to sync checkbox and finally click on the + sign to select the folder you want to keep offline.