In Exchange 2013 Cumulative Update 1, Microsoft added the ability to open public folders in Outlook Web App. Although OWA is supported, it is in a limited way... You can add and remove favorite public folders and perform item-level operations such as creating, editing, deleting posts and replying to posts. However, you cannot create or delete public folders from OWA.
To view Public Folders in OWA you have to right-click on Favorites and select add public folder:
Next select the folder you want to add anc click add:
Unfortunately, public folders can only be added one at a time, so you will have to do this for every public folder you want to add...
Once you have done this, the public folder(s) will appear under the Favorites menu.