If I move into a showdown with Diana Lunari and my opponent uses Gust, can I still use her ability?

Yes. Diana's "When a showdown begins here" trigger is already on the chain by the time your opponent can Gust her, and triggered abilities resolve independently of their source — so she doesn't need to stay on the battlefield.

Gust has [Reaction], so it can be played in response to Diana's trigger, but it only removes her; it doesn't counter the ability. When her trigger resolves, you can still pay [1], [Predict], and draw the spell if revealed — her card text doesn't require her to still be in play (FAQ #10222).