Does Overzealous Fan trigger when I defend, so it triggers before combat showdown starts? If my opponent just moves in with one unit and I use that ability, do I still win the combat even before showdown starts?

No — you don't win the combat; it's "No Result."

First, a timing fix: Overzealous Fan's "when I defend" doesn't trigger before combat. It triggers when the Fan gains the Defender designation, which happens in Step 1 of Combat when the showdown opens as a Combat Showdown (Rule 464.2.c.3) — so there's no window to use it beforehand.

If you do use it (killing the Fan and moving the attacker to its base), the battlefield ends up with zero units from either player. When the combat resolves, neither player has units remaining, so the result is "No Result" (Rule 466.3.d), and the battlefield becomes Uncontrolled (Rule 466.5.b). No conquer, no VP, no combat-win triggers (FAQ #10396).