Earlier I discovered a post on Gizmodo telling me how to download the 2.0 release and manually install it. After a harrowing 30 minutes while I feared I was going to lose my photos and notes, and another 15 minutes syncing all my music & videos, I finally got it all back to where it was. (Note: be sure to back up your phone first. If you manually update it, the backup and restore doesn’t happen. The Backup command has moved to a contextual menu on the iPhone in the left column.)
I grabbed a couple of free iPhone apps, and installed them. Then I bought a few more apps, and when I tried to sync, iTunes reported errors like:

The application "Poker" could not be installed because the application could not be found.

Huh. I figured it was because there was not enough free room, but after deleting The Incredibles and trying again, I kept getting the same error.
I finally decided to power-cycle the phone, and quit & relaunch iTunes. As I write this, I’m watching the sync progress. It seems to be working. Not sure what the actual problem was.