What are some ways to backup the vps? How should a backup be done while in the rescue image or while the vps is running?
A full backup is only possible while in the rescue image, and there is often not enough tmp space to use as a destination, so it needs to be copied over the network directly to the destination. One way is to setup a special account with its own password or ssh keys on the backup destination server, then pipe tar or gzip through ssh. This works with a filesystem image directly from or to a partition, or with tar and a mounted filesystem. (Some more details: Backing_up_disk_image_via_SSH)
Also, see Restore