I'll go with Buster Sword.

I draw strength from your life, (Ultima Weapon depends on HP)
But this strength is not free, (as HP down, strength down, MP/items spent to build it up again... etc.)
You will know me as the last, (It is an ULTIMATE weapon - so it is the last)
But it's the first that you seek. (First - initial equipment => Buster Sword)

So, Buster Sword.

If this is indeed the answer, I'd say you drew a close line there - it could easily be construed to mean Yuffie or Cait Sith's initial equipment as well. Fortunately, Buster Sword's description is "initial equiping [sic]", so that works as well.