I noticed there's a manual button to check for updates on the Live Area for games. If a game update is out will launching the game check for updates or do I have to manually check for updates? Do PSP games on the Vita auto/not-auto update just like PS Vita games?
No, your Vita will not automatically check for updates.
On the other hand, several common tasks, such as trying to download a game from the PSNetwork Store, will check for updates, and notify you that a new update is available.