Installing apt-get

of course you can do that, although it is strange that you missed that package/command ... Anyway, a manual way to do it is to download the package and to install it with dpkg, as follows:

1. grab the package from the web

   wget http://security.ubuntu.com/ubuntu/pool/main/a/apt/apt_1.0.1ubuntu2.17_amd64.deb -O apt.deb
   

2. install it with dpkg

   sudo dpkg -i apt.deb
   

2 (other). Alternative use of dkpg without sudo - I didn't know about these, included after comments-

pkexec dpkg -i apt.deb

if everything goes right, that should be enough.

Probably you may find that some dependencies are not satisfied, to deal with that you may need to create a folder (something like "apt-installer") and drop all the dependencies of the apt package there. The list of dependencies required will be shown by the dpkg output and the download process is similar to the one explained above. Last, if you don't feel confident of what you are doing, my suggestion is that you may want to do a fresh install of your system.

Cheers.