1.
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
PHRASAL VERB : V P -ing , V P prep
2.
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.
PHRASAL VERB : V P n (not pron) , also V n P
3.
see also finish 1, 2