Got a few questions in here so I'll break these out:
If a one of the hosting facilities goes down, how can he switch over to the other one?
There's several ways to do this, including database mirroring (new in SQL Server 2005), log shipping, and replication. I've recorded a podcast on SQL Server high availability and disaster recovery options at SQLServerPedia.
(Or is it just going to be a DNS change that will take 24-72 hours to propagate?)
Like the other post mentioned, you can set your DNS time-to-live numbers very slow, but the cooler method uses database mirroring. With mirroring, you can set both the primary and secondary server names in your connection string, and your application will automatically try the second server when the first one doesn't respond.
How can a single SQL Server instance have so many databases on it? FB has a completely separate database per account. I can't see a single SQL Server instance having more than say 200-250 databases on it! And I'm sure they have more customers than that.
The largest SQL Server I've worked with had over a thousand databases, and I've talked to a couple of other DBAs who have worked on systems with more than 2,000 databases on a server. It certainly makes management much more challenging, that's for sure.