You can tell ssh to not allow password authentication using:
ssh -i /path/to/private_key -oPasswordAuthentication=no admin@server_host
This will cause ssh to exit immediately if the private key does not work.
If you would like it to just exit if it is successful also, just add that command to the end:
ssh -i /path/to/private_key -oPasswordAuthentication=no admin@server_host exit