Here's what I was greeted with upon trying to sign in within the game, telling me to back out to the LiveArea (a screen the Vita displays when you click on an app's icon, with links to DLC, the manual, etc.) The Vita's built-in screenshot function was disabled while this message was up, hence the hasty phone snapshot.
's "LiveArea." The arrow (added by me) is pointing to the button to apply an update. That's a pretty good icon, right? I guessed that was the one for "update," and I was right.
Tap the "update" button and you'll get an explanation of the update, and the option to download it.
You can then follow its progress, like any other download, in the menu at the top right corner of the screen.
Having to back completely out of a game to find and apply the update is kludgey, but once you've figured that out, it's reasonably easy to use. And now that you know how to do it in advance, it'll be even easier.