How do I install libwebkitgtk-1.0-0 on Ubuntu 19.04

Based on this, it seems to only be available for up to 18.10. I would suggest to download the .deb file from the PPA here, by expanding version 2.4.11-4 of webkitgtk for Cosmic (18.10), and downloading libwebkitgtk-1.0-0_2.4.11-4.deb for the correct architecture. Then, use sudo dpkg -i libwebkitgtk-1.0-0_2.4.11-4_<your architecture>.deb. You may also need to install dependencies with sudo apt install -f.

Edit: As @fkraiem said, installing 18.04 will also solve your problems, and I personally recommend using the LTS versions unless you really want the latest version.

Install 18.04... – fkraiem