"They came first in the last soccer competition."
Do you have to use the 上一次的 construction? Could you just say 上次足球比赛？
Does there not need to be a proposition in the Chinese sentence to clarify that the action happened "in" or "at" the competition?
Not necessary. 上一次的足求比赛 is used as a topic: "As for the last football competition, they were first."