I have tried a solution that seems to work for someone else on this question:
Update table a from table b where (conditions) I can not seem to get it working, MySql
Try this:
UPDATE video_data SET date_timestamp = (SELECT video.date_timestamp FROM video WHERE video.video_id = video_data.video_id)
Although the error in your actual query is simply that you missed a "SELECT"
UPDATE video_data SET video_data.date_timestamp = SELECT video.date_timestamp FROM video_data JOIN video ON video_data.video_id = video.video_id
But I don't think it is what you want it to be.
Try this syntax out:
UPDATE video_data, video
SET video_data.date_timestamp = video.date_timestamp
WHERE video_data.video_id = video.video_id
UPDATE video_data, video SET video_data.date_timestamp = video.date_timestamp
WHERE video_data.video_id = video.video_id