Try the following:
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 simple in that you missed "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 do not think that this is what you want.
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