Okay.... so. I think this is right, but just checking:

Boba Fett:

After performing an attack, if the defender was dealt a faceup Damage card, you may discard this card to choose and discard 1 of the defender's Upgrade card.

Integrated Astromech

When you are dealt a Damage card, you may discard 1 of your Astromech Upgrade cards to discard that Damage card (without resolving its effect).

If I take a face-up damage, I junk the Astromech to discard the card. Hence Fett can't be used?

The Astromech effect takes place during the attack (when dealt the damage) so if I have it right, the discarded card no longer 'exists' for Fett to trigger off, since he only applies once the attack is completed.

Or does the fact it was dealt, even if it's not longer around, count?

No, he still dealt you the card. That's what matters, not what happened to it afterwards.

So, if he dealt you a faceup card, you would IA it away, but he could still discard one of your remaining upgrades - if you had any left...

As written, Integrated Astromech would trigger first removing the critical damage, but Bobba could still discard other equipped upgrades from your ship, since even if you discarded it later, you still have been dealt a faceup card, which is enough to activate Bobba.

If two times isn't enough here's the third: Boba is triggered when the card is dealt. What happens to the card afterwards is completely irrelevant.

Also see:

Boba vs. Chewbacca (either one)

Boba vs. Determination

Boba vs. Draw Their Fire (although in this case Boba does NOT work because the target may not be the one drawing the card)