LTM Upgrade Process

Since you're already at 11.1, you should just install 11.3 to a separate volume and reboot into that volume. By default your current configuration should follow the installation. If you've been on this build for awhile, make sure to renew your license before upgrading. The back out process would involve booting back into the 11.1 volume.

Also, for an Active/Standby configuration does it make sense to upgrade one side, let it run in production for a day or so, then finish the other side?

Always a good idea in my opinion.