I assume you have nix installed as a package manager at least if not the whole NixOS thing and a working home manager setup.
Most of the setup below was obvious, but the password stored externally to your configuration wasn’t quite defined in the section on setting up external sync services. Using feedbin-passwordfile you can set the location of the file you store the password in. Then that file only contains the password on the first line as described in the Newsboat password API docs.
Unlike some other applications, Newsboat will not work if you use single quotes around the extraConfig options so you have to escape " to stop the parsing error.
programs.newsboat = {
enable = true;
autoReload = true;
reloadTime = 10;
extraConfig = "
urls-source \"feedbin\"
feedbin-login \"curtis@curtismchale.ca\"
feedbin-passwordfile \"~/.newsboat/feedbin-password\"
";
};