Those menu items are rendered by the Aikau "alfresco/menus/AlfMenuBarItem" which does have support for adding icons, however the default icon position is to the left and not above (I suspect you're trying to recreate the look of the old 3.4 header?)
Firstly you're always going to need to extend the header (you're going to make sure you follow the steps in this SO question and answer I created to ensure that the customization is applied to all pages in Share. For general guidance on extending the Share header you can read this blog post.
You're going to want to either extend the AlfMenuBarItem widget (or create an entirely new widget) and then update the page model defined by the share-header.lib.js file - in particular the part of the model created by the "getSiteNavigationWidgets" function to swap out the default widgets with your custom version.
You can do this by searching for the IDs of the menu items... the steps described in this blog post will help you get the IDs and snippets of code for finding them.
Hopefully that makes sense - please reply if you need more assistance with this!
