SQL Server 2012 SP1 to SP3 with Mirror
Is there a reason you don't want to failover? 

That would be the most common way. To do a rolling update and away you go.

If you have a witness, disable it first.  

When you patch a mirror in this way the mirror will continue to function, to a fashion, and you'll have no application downtime.

When a mirror partner is patched it will leave the mirroring session, but then rejoin it when it is complete - then re-sync.  

You can do it the way you described but you're going to have downtime when you patch the prod server and I'm guessing the patch when applied to Prod will trigger failover if that's how your mirror is configured.