If an opponent starts a showdown at a battlefield I have a unit at, I react by playing Heedless Resurrection and get back Overzealous Fan which I play to my battlefield, can I then use Overzealous Fan's effect to move the opponent's unit to base?

Yes, you can use Overzealous Fan's ability to move the opponent's attacking unit to its base.

Sequence of Events

1. Showdown Begins: Your opponent moves a unit to a battlefield you occupy, applying Contested status. Per Rule 345, the opponent (the player who applied Contested status) gains Focus.
2. Reaction: While the opponent has Focus, you play Heedless Resurrection as a [Reaction]. This creates a chain, making the state CLOSED. You kill a friendly unit as a cost to resurrect Overzealous Fan.
3. Fan Enters Play: Overzealous Fan enters the battlefield.
4. Chain Resolution: Heedless Resurrection finishes resolving.
5. Combat Cleanup/Defender Designation: Following the resolution of the Heedless Resurrection chain, the game performs a cleanup. Overzealous Fan gains the "Defender" designation because it is now present at the contested battlefield (Rule 459.2.b.4.a).
6. Fan's Trigger: Because Overzealous Fan gained the "Defender" designation, its "When I defend" ability triggers and is placed on the chain.
7. Resolution: When the trigger resolves, you may choose to pay the cost (killing Overzealous Fan) to move the attacking unit to its base.

Why this works

• Defender Designation: According to Rule 459.2.b.4.a, if a unit controlled by the defender becomes present at a contested battlefield after the initial combat setup, it gains the Defender designation during the cleanup phase following the action that caused it to become present.
• Trigger Timing: Overzealous Fan's ability triggers specifically upon gaining the Defender designation. Even though this occurs mid-showdown, it is a valid trigger for the ability.
• Cost Payment: Per RiftJudge FAQ #9047, the cost of killing Overzealous Fan is paid at the time of resolution. As long as Overzealous Fan is on the battlefield when the ability resolves, you can pay the cost to move the attacker, even if the Fan has moved or been swapped in the meantime (provided it remains on the battlefield).

Important Note on Focus: You correctly noted that Rule 345 assigns Focus to the player who applied Contested status (the opponent). While the opponent has Focus, you are permitted to play [Reaction] cards to respond to their actions.

Note: I could not fully verify this ruling against the retrieved rules, specifically regarding the timing of when a non-active player may initiate a chain during a Showdown Open State.