If you finish up in a particular place or situation, that is the place or situation that you are in finally.

You'll finish up dead if you carry on drinking like that. 如果你继续这样喝下去的话，会把自己喝死的。

She married a Spaniard and finished up in Barcelona. 她嫁给了一个西班牙人，最后在巴塞罗那定居了。

to eat or drink all of what you are eating or drinking

Finish up your dinner and you can have dessert. 把晚餐吃完你就可以吃甜点了。