Importing an ssl on .CRT format,, I have passphrase but not the private key

I have a customer that owns a domain, they will change their DNS so it points to my server so I host their secure site.
They got an SSL certificate from GoDaddy (they generated the CSR with a server of their own and used a passphrase).
They picked the certificate at GoDaddy (a .CRT file) and now they sent it to me, on the email they wrote the passphrase. They say that with this, I should be able to import it on my server.

They way I understand, I need the private Key, basically, they have to install the SSL on their computer used to generate the CSR, and then using the pass-phrase they should export the certificate and the private key so I can use those to import the cert on my server.

They insist that using only the file they sent and the pass-phrase I should be able to import it, using openssl or some other utility.

I open the .CRT file they sent on Notepad++ and I only see the block for the Certificate, not the Private key.

Im confused, do I need the private key or not in order to use that .CRT file on my server and host their site?

Thanks!!