Does Diana Lunari trigger resolve if she gets gusted to hand in response?

Yes. Once Diana's "When a showdown begins here" trigger is on the chain, Gust only removes her — it doesn't counter her ability, and triggered abilities resolve independently of their source (FAQ #10222; Rule 392).

Gust has [Reaction], so your opponent can play it in response to the trigger, but when her ability resolves you can still pay [1], [Predict], reveal, and draw a spell if revealed — her card text doesn't require her to still be in play.