Been doing a bit of fiddling around and you can actually back up quite a lot with built-in software and ES File Explorer.
Asus' own App Backup can back up the application files (.apk's) and/or the settings for most apps to a single encrypted file. It's a bit limited in that it backs them up to internal USB storage so you have to have enough room there, but it also copies the backup to external storage (e.g. Micro SD, SD or USB stick) if you have it. It backs up to a single file with a limit of 2GB so I had to split mine into two backups.
ES File Explorer has a function called 'App Manager' which lets you back up the individual .apk files to wherever you like (although there are a small number that it doesn't seem to be able to back up).
Neither of these methods backs up the huge additional downloads that some games etc. require, so I used ES File Explorer to copy my whole 'sdcard' folder (which is the internal USB storage) to a folder on my SD card.
Hopefully, with all of that plus Android's built-in backup syncing I should be OK.
Time to give it a go ...