Copy EVERYTHING to backup server, change DNS ip's to reflect those of backup server, and within 24-36 hours of updating IP's the sites would be at new server with no downtime.
You would probably lose a little mysql (etc, someone posts on forum while they are resolving to busy server, but you copied the mysql to backup a few hours before, that post wont be on the backup server)
maybe use rsync for data, setup a mysql server on the backup and redirect all queries to it after copying whole content, so the mysql db wont lose too much, then when fixed box is back up, sync the db content (as good as possible) and redirect all mysql queries to it
Don't bother with DNS changes if the backup server is right there on the same subnet. Just alias the IP of the server going down on the backup server and remove the alias when the primary server is back online. The switchover will be instant. The only problem is usually a few seconds/minutes of database blackout as sql is copied between servers, but as said above, you can use rsync and change the DB host to virtually eliminate db downtime.
If your backup is on another network... it'll take some extra time, but you can still keep "downtime" (again, DB only) to a minimum using short DNS TTL.
If the servers are in the same Datacenter, you could do some tweaking and transfer IP's after a HD swap. If not, there is 72-96 hours of downtime. The Nameservers have to propogate, then after that - the domain names.