Sure you can. Sarcasm is essentially saying the opposite of what you mean, but in a certain tone which reveals your true feelings. There's no reason you can't employ win in this fashion. Winning something has positive polarity (no or not doesn't appear in the sentence) but it doesn't entail positive emotions (the result might make you feel bad).
Your example I won lots of downvotes does just that. You've "won" (been awarded or successfully acquired) something you don't want, so the outcome is undesirable.
You might also use win sarcastically to describe a hollow or meaningless victory:
I heard you won the tournament, great job!
Oh yes, I won a crushing victory... because nobody else showed up to compete.
The implication here is that the second person didn't really win at all as there was no competition. There's a negative connotation because joining a tournament and winning by default tends to make one quite cranky.
Someone extremely cruel might say you won! very sarcastically to someone else finishing in last place or performing exceptionally poorly. Similarly, someone doing something trivial or stupid and acting as if they had made a great accomplishment might receive a dry remark of congratulations, you win the prize.