Even if later backups will be relatively quick, I'm still concerned that the first backup will take too long over my network connection. Is there a way to do it faster?

If there is a lot of data or a slow network upload speed or both, making the initial backup take too much time, we can ship out an external hard drive to be connected up to the system that is to be backed up. The initial backup can be done from that system to the hard drive, then the hard drive can be shipped back to us and we can upload the initial backup from the drive.