Can I use the 1 energy produced by Diana, Scorn of the Moon during my opponents open showdown. For example to use it to play "Gust" against their unit before it conquers?

Yes — that energy is spendable during your opponent's showdown, and you can use it to play Gust before the conquer resolves.

Scorn of the Moon's restriction is only "Spend this Energy only during showdowns" — it doesn't care whose showdown or whose turn it is. Activate Scorn of the Moon at Reaction speed (its ability has [Reaction], and Add abilities can't be reacted to), which puts 1 energy in your Rune Pool. Gust is a [Reaction] spell, so it's legally timed in a Showdown Open state when you have Focus (and even in the Showdown Closed state while a chain exists). You can also activate Scorn of the Moon directly during step 4 (pay costs) of playing Gust (Rule 357.1.a).

Two timing notes: you must play Gust while the showdown is still open — if both players pass and the showdown closes, the opponent establishes Control/Conquers (Rule 348.2) and there's no reaction window to stop it after the fact. And if the showdown ends with the energy unspent, it can't be used outside a showdown and will be lost when the Rune Pool empties (Rule 167).