#Create a new keypair: (go with the defaults)
ssh-keygen

#Copy the public key to the server: (password for the last time)
ssh-copy-id user@my.server.com

#From now on the server should recognize your key and not ask you for the password anymore:
ssh user@my.server.com

visudo 
username ALL= NOPASSWD: /sbin/shutdown

#Script zum Aufruf:
ssh -tt username@pcname 'sudo shutdown -h now'