It's the start of my turn and time to activate my hero. I have 1 HP remaining and I'm poisoned. I fail my test and suffer 1 damage and this knocks me out.
My question is, since I haven't taken any actions yet can I do a stand up this turn as my only action or do I have to wait until next turn to do so?

If you are defeated at any point while it is your turn, your turn ends. You cannot stand up until the next round (another hero could revive you during the same round, but you couldn't take a second turn, as yours has already ended).