If you finish up something that you have been eating or drinking, you eat or drink the last part of it. Finish up your drinks now, please. see also finish 1, 2
finally be or do something; "He ended up marrying his high school sweetheart"; "he wound up being unemployed and living at home again
If you finish up in a particular place or situation, you are in that place or situation after doing or experiencing several things. They had met by chance at university and finished up getting married He's probably going to finish up in jail for business fraud. = end up, wind up
finally be or do something; "He ended up marrying his high school sweetheart"; "he wound up being unemployed and living at home again"