There is huge amount of free mail service, unless you are talking about your own domain.

Even with custom domain, there are some mail providers that accept Bitcoin. See here: https://www.privacytools.io/providers/email/

And, once more: my previous point was that even if providers know your id, it is not public. There is no service you can use to ask what is name of e-mail address owner (I'm ignoring "whois", which is domain-specific, not mail).