At the time writting, It looks like the Search Result Alert Me functionality doesn’t work in Office 365 (I’m using the E3 plan).
For instance, I’ve setup a Search center with the default settings : (here is my result.aspx page)
When I edit the result.aspx page:
When I edit the Search Results web part, I get the web part properties; select the Settings group of properties and you will find the Show AlertMe link :
Edit the Result page again, edit the web part and you will notice that the Show AlertMe option stays unchecked. Annoying. I’ve reported this to the Product team.