Best practices for updating Exchange Server 2013 Standard to CU13 from Exchange Server 2013 SP1

Our company is looking into possibly migrating to Office 365. I currently have two Exchange 2013 Standard servers running on Windows Server 2012 R2 configured with a DAG. In order to establish a "hybrid" configuration, I need to update the Exchange 2013 servers to CU 12 or 13. Embarrassed and ashamed as I am, no previous CU have been applied to the servers. This is partly due to my difficulty of getting approval to perform server maintenance on these particular servers. Once I verify my backups have finished. My plan is to move all active databases to one server and configure mail flow to go to the first server, begin updating the second server by applying all available Windows Server patches, restart and verify all services are working. Being fairly novice to DAG, my understanding is I need to put the Exchange server into maintenance mode. Once done, apply CU 13. Restart the server and verify services. Once complete, exit maintenance mode, configure mail flow to point to the second server, activate the databases to become active copies. Repeat the maintenance/patching process on the first server. Does anyone have any recommendations or see any objections to this procedure? Any best practice notes would greatly be appreciated. Thank you for your time.