When you work with Sharepoint in Office 365 without configuring Exchange online you will get an error message when you try to set alerts.
To solve this issue you need to follow these steps:
Open the Office 365 Portal in the admin page.
Select "Manage" under SharePoint Online
Select "Manage User Profiles"
Under "People" select "Manage User Profiles"
Search for a user that you would like to assign an e-mail
Select the item-menu and select "Edit my Profile."
-If you go over the username with the mouse, you will notice that a small black triangle appears at the right side of the username. Select that and you will see the "Edit my Profile" option or select the user name and you will see these options as well.
Look for "Work e-mail" and add your e-mail address.
Select "OK" in order to make the changes to apply.
This change may take 24 hours to get synchronized. Please give it a day.
Try to manage alerts again.