Based solely on the rules in the rulebook and on the card, I would be inclined to agree with Flat. However, the FAQ has provided an answer which suggests you cannot do this:
Q: Can a knocked out hero do anything after standing up during his turn?
A: After performing a stand up action, the hero’s turn immediately ends.
This means the hero cannot suffer any fatigue to gain movement points
or use any other skills or abilities.
Granted this doesn't explicitly mention potions, but it does expressly forbid doing things that aren't actions (ie: spending fatigue to gain movement.) You stand up, then your turn ends. That is all.
Mind you, an adjacent hero could use a vitality potion on you, even before you've stood up, so a little teamwork can still help ensure you have full fatigue after standing.