Just to make it clear, you're not having trouble running npm-windows-upgrade to upgrade your npm version. You're having trouble installing the upgrade tool.
First, make sure you are running PS as an administrator. If that doesn't help, you should probably look at the section fix-an-attempted-upgrade from their readme.md:
Fix an Attempted Upgrade
Chances are that you attempted to upgrade npm before, it somehow failed, and you then went looking for this tool. If the tool fails to upgrade, it may be troubled by partial changes done during npm install npm or npm upgrade npm. In that case, you will have to completely uninstall Node:
- Uninstall Node.js (select
Uninstall, not the Repair option). - Go into
%programfiles%\nodejs and delete the entire folder. - Delete
%appdata%\npm and %appdata%\npm-cache. - Edit your
PATH and remove everything that references npm (to do so, hit "Start" and search for "Environment Variables"). - Reinstall Node, then install this tool - and only use this tool to upgrade npm, do not attempt to run
npm install npm.
PS: There's also a chance that you're running an NPM version that is too old be be compatible with installing npm-windows-upgrade, but I'm saying that based on nothing.
npm --version?