If I hold Reckoner's Arena with Yone, Blademaster, will he do damage?

No, Yone, Blademaster will not deal damage in this scenario.

According to the official rules (FAQ ID 9922):

1. Reckoner's Arena acts as a bridge that forces the "When I conquer" abilities of units present to trigger during a "Hold" action.
2. However, this does not bypass the conditions written on the unit's ability. Yone, Blademaster's ability specifically requires that the battlefield was "previously uncontrolled" for the effect to trigger.
3. Since a "Hold" action by definition occurs at a battlefield you already control (Rule 464.2), the condition that the battlefield was "uncontrolled" is not met.

Therefore, even though Reckoner's Arena activates the "conquer" event for the purpose of checking triggers, Yone's ability will not successfully trigger because its specific condition remains unfulfilled.