So finally I have decided to move my blog to Office 365 and first thing I would like to do is to thank Martin Hatch. When I started playing with my blog site I realized that there is no way I can turn the anonymous permission on for this site. Luckily I found proverbially this article by Martin Hatch which shows how you can turn anonymous permission on for your blog.
So if you want to host your blog on Office 365 and you are running into issues with anonymous permission setting then I would highly recommend you to check this article out.
Over the next few days I will be moving all my old blog posts and then continue to blog about some interesting things I learned with SharePoint.