How to upgrade SQL Server 2008 to SQL Server 2014 servers with mirroring configured
Your research is correct you have to first apply service pack on mirror instance and then failover the mirroring so as to make this mirror instance as primary. Now you have to apply patch on instance which is nor mirror(which was previously primary). I have done it many times and I do not see much Gotchas.


  and also how should I proceed with this upgrade without causing an outage?


There will always be an outage you just cannot avoid it, when you failover SQL Server have to be stopped on current instance and failed over and started on other so you can see outage is just unavoidable.

For your reference


Minimizing downtime when patching instance which is mirrored
Upgrade Mirrored instance
Steps to Apply a Service Pack or Patch to Mirrored SQL Server Databases


Start reading the first 2 articles and then the third one has step by step method with screenshots. 

The high level steps would be


Remove the witness server from mirroring(if you hjave configured witness)
Apply the service pack to the mirror server
Failover the mirrored databases over to make it principal
Apply the service pack to the former principle server (which is now a mirror)
Apply the service pack to the witness server
Add the witness server back to the mirroring configuration