Automator: back up certain folder on specific disk, whenever its connected

Create a folder action and attach it to the /Volumes folder (this is an invisible system folder used to access the file systems on mounted volumes). It will be triggered when a volume is mounted, so just filter for the desired name and then perform your backup.

The /Volumes folder is normally invisible, so a hoop needs to be jumped through in order to be able to select it in the file dialogs. In the the Finder's Go > Go to Folder... menu item, type in /Volumes and go to the folder - this is the hoop, you can just close the Finder window unless you want it for something else.

Open Automator, choose the Folder Action template, and select Other... in the workflow popup menu to get a dialog to choose the folder to attach it to. You won't see the /Volumes folder in the dialog, since it is invisible and all, but you just jumped through that hoop, so it will be under Recent Places in the toolbar's path popup menu.

Now you can use a Run AppleScript action to run a short script that checks the name of the volume that triggered the folder action (any disk/image mounted is added to the folder). If one you are interested in is there you continue with the backup/sync, otherwise the script cancels the rest of the workflow. Something like:

on run {input, parameters}
  set backupNames to {"name1", "name2"} -- names to look for - first match wins
  tell application "System Events" to repeat with theVolume in the input
    if (get name of theVolume) is in backupNames then return theVolume
  end repeat
  error number -128 -- user cancelled
end run

Follow the script with whatever actions you were using before before.