In order to configure Windows 8 mail app using the exchange activesync you need to:

1 - install the zimbra certificate
2 - export and install the zimbra server ca
3 - configure the server field with the hostname associated to the certificate, tipically the hostname of the zimbra server

Follow this steps to export the ca:

cd /opt/zimbra/ssl/zimbra/ca
openssl x509 -in ca.pem -inform PEM -out crt.cer -outform DER

Now install the exported crt.cer to your PC.
I don't know if this is working for imaps and pop3s too.