Facebook update on my iphone

Your question doesn't really make sense.

Facebook has no connection to an Apple ID. Facebook has its own user name and password.

It would help if you told us exactly what steps you are taking, and exactly what request you are seeing.

For example, are you going to Settings > Facebook? There it asks for a user name and password (which has nothing to do with an Apple ID).

Ah, OK, your explanation helps, thanks.

First, are you properly logged in at Settings > iTunes & App Store?

Second, is the Apple ID that is signed in there the same one that owns the Facebook app?

Assuming that you are logged into Settings>iTunes & App Store using your Apple ID, you'll need to delete the Facebook app from your phone and redownload. Apparently, it was originally downloaded using someone else's Apple ID.

These steps should have been taken before the phone was turned over to you:

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

If they weren't you're likely to have problems. Who's Apple ID is showing under Settings>iCloud? If it's not yours, get in touch with the previous owner and follow the instructions above.