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.
This has been resolved as per the FAQ page 7 under spells:
Q: Can a delayed investigator in an Other World cast Find Gate?
A: No the only things a player whose investigator is
delayed in an Other World may do during his Movement
Phase is stand his investigator marker up and/or trade with
another investigator in the same Other World area.
This answer previously held links to places where the rules around the Find Gate spells was discussed. Since the final ruling from Fantasy Flight Games i have removed this information and the links.