Spells (and other items) will usually have some casting phase or timing defined on the card that states when they're allowed to be used.
Find Gate has Movement: on it, so you can only cast Find Gate during the Movement phase. So if you were drawn through a gate due to an Arkham Encounter that said some thing like 'A Gate and a Monster appear!' you wouldn't have the opportunity to cast Find Gate before your first Other World encounter. But you would be able to cast it the turn after that.
Additionally, Find Gate has been subject to some scrutiny due to the fact that the delayed status doesn't explicity prevent you from casting spells, only from using movement points. So technically it's allowable that a Mythos Card draws your investigator through a gate and you use Find Gate before you have even a single encounter. Here's a thread on the Fantasy Flight forums where people talked about it:
Also a thread on BoardGameGeek where people talk about it:
So currently the preferred way of playing these situations is that you're supposed to have at least one encounter in an Other World before you're able to use Find Gate, as being able to get through an Other World for essentially free is against the spirit of the game. So become un-delayed takes your entire movement and the combat penalty from the Nightgaunt immediately ends your movement phase. But again, there's nothing in the rules that explicity states this is the way it has to play out, so play it out however you like.
As a second example of timing, Astral Travel (from the Curse of the Dark Pharaoh expansion) has a timing of Arkham Encounters so you couldn't use it to rush into a group of monsters, take one of two out and then Astral Travel out of there.