The concept of Content Type publishing became a possibility in SharePoint 2010 using a new feature called the Content Type Hub. In this scenario you could create a site collection, activate the content type hub feature, and then publish content types to all site collections. There are some issues and limitations with this concept and this blog outlines them well. Also if you are want some more detail about the content type hub itself and how to use it, this blog has some great information! So knowing this feature is available in 2010 and 2013 is this same feature is available in Office 365?
Yes! The Content Type Hub is already built for you in Office 365! Do not create your own site collection with the intention to use it as a content type hub as it will not work when you try to configure it. The content type hub feature is under the URL "/contenttypehub." So in your tenant you want to use the URL:
Unfortunately you will not be able to see this site collection in the SharePoint Online Admin center to manage it, but it really does exist. Older tenants actually may still be in the 2010 format and need to be upgraded. If no one has access the site collection yet to set security, use a tenant global administrator account to grant the proper access.
Once you have access, you can publish content types the same way you would in an on-premises situation!