WinSCP supports SSH keys.
In /etc/ssh/sshd_config set
Copy your public key to /root/.ssh/authorized_keys (there will be a similar file inside of ec2-user's home directory). If you want to just copy that type
sudo mkdir -p /root/.ssh
sudo cp /home/ec2-user/.ssh/authorized_keys /root/.ssh/
This way you can login as root but only with ssh keys. If you have already assigned a password to root, you can disable the password on the account by typing:
sudo passwd -l root
You will also need to reload ssh:
sudo service sshd reload
Assuming you have already converted the amazon public key file to putty format, point WinSCP to that private key file and you should be able to login without a password. If you have not converted the key file you can use puttygen to do so.