This is a Step by Step tutorial on how to add Public or External contacts in Lync Online. To add user from External domain you first need to add that domain in your Lync online as Allowed connection. Lets see the detailed Steps.
1. Login to Office Admin Screen and under Lync click on Manage.
2. Next In Lync Online Control Panel click on External communications.
3. Next Enable Domain federation mode.
4. In the popup select “Turn Off External communications with all except allowed domains.
5. Next to add external users in Lync we would need to add their Domain as Allowed connection. Click on “add a domain” and enter the domainname of your external contact.
6. Next verify that the Domain name is added as “Allowed”.
Now you can add external users as contacts from the above added domain in your Lync Online.