This is up to the DM
The pedantic answer is that spells and effects only do what they say. This means catapult only deals damage to the flask, not activating its features. You have not taken the action nor made the ranged attack with it that it describes. How many hit points any one object has is also up to the DM, but using the suggested hit points from the DMG (p. 247) a resilient tiny object has 2d4 hit points, so it is reasonable to assume the flask is destroyed. However, the alchemist's fire has no description for what happens when it is destroyed, so it is on the DM.
Your DM might use the base description as a basis for what should happen, and apply the fire damage to the target. Or, they may rule that it does not hit in a sufficiently focused manner to apply harm to the target and so the target does not suffer damage. The main perceivable argument for not allowing this, is not setting a precedent for applying real-world logic to the game rules, which might benefit some characters and their abilities much more than others and/or eat up too much time at the table with arguments. On that note, as will any case of a strategy having ambiguity, ask your DM ahead of using it (possibly outside of game time) on how they would rule, so you know that before using it.