One of my favourite songs was 70th the last time they did the rankings, but it moved up 4 spots to 66th this time.
この場合、次のような言い方ができますよ。
ーOne of my favourite songs was 70th the last time they did the rankings, but it moved up 4 spots to 66th this time.
「わたしの好きな曲は前回のランキングで70位だったが、4つ順位を上げて今回は66位だった」
ここでは、to move up four spots で「4つ順位が上がる」と言えます。
ご参考まで!