I would like to try the program Billreminder 0.4.0. But I need a little help with installing it. There doesn't seem to be a .deb package available in the software center or Synaptic.
You seem to be right. There is no packages in Ubuntu and they don't appear to have any available for download either. That means you'll have to do it yourself. It may sound complicated, but it really isn't. You will have to read a couple of pages, though: https://help.ubuntu.com/community/CompilingSoftware
Basically: (the above page explains all of this in detail)