Are you asking about doing message tracking? You can do that from exchange system manager. If you are asking about opening mailboxes on the exchange server, that is not an option. You CAN exmerge a mailbox out to a pst. But I do not know if that is what you are asking. Please provide more details thanks.
Message tracking is what you need for your purposes. It is turned off in Exchange by default because of higher system requirements. To enable message tracking, open Exchange System Manager. Navigate to your server. Right click on the server and click on properties. In the General tab, Check Enable subject loggin and display and Enable message tracking. You can set the log location and log removal settings here as well. Do the same for all exchange servers in your environment, and no reboot is required.
In the Exchange System Manager -> Tools. You will see Message Tracking Center. You can search for the user and see the logs for every email that user sent and/or recieved along with the size.