Does Purple Ezreal pay for the cost of Diana Lunari?

No, Ezreal, Prodigy’s ability does not apply to the cost of Diana, Lunari’s triggered ability.

According to RiftJudge FAQ #286, #655, and #1354, Ezreal's cost reduction specifically applies only to optional additional costs—costs paid when playing a card that explicitly use the terminology "as an additional cost" (such as Accelerate or Repeat).

Diana, Lunari's ability is a triggered ability ("When a showdown begins here...") with an optional cost component ([1]) that you pay to resolve the effect. This is not an additional cost paid when playing a card, and therefore does not qualify for Ezreal’s discount.