1.no sooner ...than句型可以表达“刚刚.就.”
2.no sooner放句首,前面要倒装
3.“到达故宫”这个动作在“开始下雨”之前,而"开始下雨"用的是是过去时,所以"达故宫要"要用过去完成时.
因此应该是
No sooner had we reached the The Palace Museum than it began to rain
希望能帮到你!