How to upgrade SharePoint Site Collection on an Office 365 P service plan
If you have a customer on P1, and they still have the old SharePoint 2010 site, you can manually upgrade the site collection if they are ready. You do not have access to the SharePoint Admin Tenant to see the list of Site Collections so the process is a little bit different.
*P Plans have been discontinued. This article may not be up to date
The first thing you want to do is log into the SharePoint Site, then on the top left, click the Site Actions drop down.
Next, you want to click on Site Settings
On the Site Settings page, under the "Site Collection Administration", if all you see is Go to Top Level Site Settings, click that, if you see more, proceed to the next step.
Under the Site Collection Administration, you need to click on "Site Collection Upgrade"
Once you click the Site Collection Upgrade, it will take you back to the Main page of your SharePoint site, but in the middle of the page, you should now see a blue box that you need to push to start the upgrade. Once you push the "Upgrade this Site Collection", the upgrade will automatically start. It should take less then 10 mins. Once its complete, you will have the new SharePoint 2013.