ffmpeg -i out_of_sync.mkv -itsoffset 0.5 -i out_of_sync.mkv -map 1:v -map 0:a -c copy synced_fixed.mkv (Adjust 0.5 seconds as needed—positive or negative.)

For more FFmpeg help, run ffmpeg -h or consult the official documentation. And remember: the only thing more challenging than a Bushtucker Trial is debugging an incorrect -itsoffset value.

To save storage while keeping 1080i quality (typical for 2013 HD broadcasts):