OpenSSL Certificate Renewal with same keys and NO CSR

So how do I go about using OpenSSL to accept the new .cer file and associate it with the existing private key? I have all the relevant passwords/passphrases, etc.

You don't need to "associate" the new certificate with the existing private key because it already is. The CA simply used the public key in the existing certificate (where your private key is for) for the new certificate. And since the public key is still the same even if the certificate has been changed the private key is the same too.