Sony still hasn't announced a date for PSOne Classics to appear on the PlayStation Vita, but we at least know the system update number. Sony will be adding the support with firmware 1.8, said to be coming sometime this summer. Currently, the system is at firmware v1.69.
We don't know what games will be coming once the functionality has been added, but we do know that games will come in waves. Rather than make the entire Classics library (already on PS3/PSP) available instantly, Sony will be releasing them in regular updates. If you already own the game on PlayStation Network, you'll be able to download it to your Vita again for free.
Sony showed Final Fantasy VII, Tomb Raider, Wild Arms, Twisted Metal 2, Arc the Lad, Cool Boarders 2, and Jet Moto when it announced the Classics at E3, so it seems those are probably set to come earlier than later.