Yes, the patch should get installed automatically on the replica. To confirm a patch is installed on the replica you would monitor and review the log files on the primary.
In the patch log on the primary, it should display "Required Backup will begin shortly", navigate to the backup log when the process begins. In the backup log on the primary you should recieve a the results " The backup operation has completed".
Once the backup has completed, navigate to the primary log. This log should display multiple patch related files being copied to the replica’s IP address. The last patch related file that should be copied to the replica is the file parpack.replicate. Once this file is copied, the next file that should be copied is the backup file. Once the backup file is copied to the replica's IP address, view the primary's patch log. The last result should read "Tha patch was successfully applied to the system". In the case of an HA pair, this message indicates the patch applied successfully on the primary and has been successfully copied to the replica.
To verify the patch is properly installed on the replica, monitor the replica log and verify the replica receives par pack instructions, as well as the full backup.