There is no indication that the bottle came from the cauldron at all. Even though there were four prepared cauldrons upon entering class, Slughorn pulled the prize out of his pocket:
"One tiny bottle of Felix Felicis," said Slughorn, taking a minuscule
glass bottle with a cork in it out of his pockent and showing it to
them all. "Enough for twelve hours' luck. From dawn till dusk, you
will be lucky in everything you attempt.
He is not shown filling the bottle, so there is no reason to believe that the potion in the cauldron is finished. Also, considering that the Veritaserum is still boiling, it seems likely that none of the potions are actually complete at this time:
He indicated the cauldron nearest the Slytherin table. Harry raised himself slightly in his seat and saw what looked like plain water boiling away inside it.
Also, specifically when talking about the cauldron of Felix Felicis:
"Sir, you haven't told us about what's in this one," said Ernice Macmillan, pointing at a small black cauldron standing on Slughorn's desk. The potion within was splashing about merrily; it was the color of molten gold, and large drops were leaping like goldfish above the surface, though not a particle had spilled.
Given the size and location of the bottle, and the fact that he had at most a couple of months at the school, it is most likely that the prize was from Slughorn's personal stash, and the cauldron on display was a batch he began preparing to restock.