MYSQL：前と次のビデオIDを取得するためのクエリ？

次のようなものを使用できると思います：

SELECT * 
  FROM videos AS c 
 WHERE (VideoId = (SELECT MAX(VideoId) FROM videos WHERE VideoId < c.VideoId AND IsActive = 1)
    OR  VideoId = (SELECT MIN(VideoId) FROM videos WHERE VideoId > c.VideoId AND IsActive = 1))