SharePoint and Project online has built in outlook sync feature available. To be able to sync your tasks, your mailbox needs to be hosted on an Exchange 2013 server deployed in the same topology.
If you are using Project online on Office 365 then you need to make sure to use Exchange on Office 365 as well to take advantage of this new feature.
Following is the example:
Below is the list of tasks available under My Tasks page in PWA
To sync it with outlook select Task tab and click on Sync to Outlook button
Click on OK to sync Tasks with Microsoft Outlook
The tasks will now become available under Task section in your Outlook.
You can update and edit the task progress in outlook and the update will show up in the Project Manager Approval page to approve.