Update: 29.04.2013: Use the out-of-the-box blog with SharePoint 2013
When you want to have your on blog with Office 365 then normally you have to go around some major pitfalls. This blog posts describes how to avoid those and to enable anonymous users to access your articles.
Update 24.04.2013: Newer Office 365 tenants have to use: http://www.fiechter.eu/Solutions/Wsp365.Anonymous.2013.zip
After the installation do the following steps
Moderation can be optionally activated by the feature Wsp365.Moderation. When you activate this feature, then users can create anonymous comments and you have to approve it. Otherwise no one can see the comments. This helps against comment spamers.
To be notified when a comment was created, activate the notification:
When you are not logged in, you don't have the ribbon on the blog. This is for better performance on mobile phones.
To get the ribbon back, just login to the member site, and open the blog again.
Access with a logged in user