The city at one time must have been prosperous, for it enjoyed a high level of civilization.
【翻译】这座城市肯定一度很繁荣，因为它曾拥有高度的文明。

at one time 虽是介词词组，但它是副词用法，意为“曾经；一度”，跟 once 同义。其位置应该很灵活，句首，句中，句末均可。放在be 后也可以，见后面的例句。

Alaska was at one time connected with the tip of Siberia.
阿拉斯加曾经和西伯利亚的东端连在一起。
He was at one time so devoted to piano playing.
他曾经那么沉迷于弹钢琴。
My younger sister was at one time a boarder.
有段时间我妹妹是寄宿生。

must have been 表示对过去十分肯定的推测，没有第二种意思。不容怀疑。