I manage to make it by installing HomeBrew first you can get this at this link: https://brew.sh/
You will have to type this before you start using brew command export PATH="/opt/homebrew/bin:$PATH"
After that type in brew update
Then after you successfully updated go for brew install node
When you have done all the stuff above you can try npm ...
You are running into permission issues. MacOS is a Linux based operating system. Linux based operating systems require administrative privileges to make editorial changes to root directories. On MacOS you simply need to run with the sudo command:
sudo npm install --g firstname.lastname@example.org
We recently changed some aspects of the build system. Could you please open an issue on https://github.com/ethereum/solidity? Also your screenshot does not really show the error message, it should be a little further up.