If you want to communicate with a priest (or anyone else, for that matter) anonymously in order to seek spiritual (or other) advice, consider using Tor Mail:

(Just Google.)

It's free and completely anonymous, but you will need the Tor Browser (also free) to access it:

(Just Google.)

When using anonymous email, do not give out personal details (not always necessary for spiritual direction), as the Site Administrators (who are also anonymous) can certainly read your emails if they want to (unless, of course, you are encrypting them), and do not use it (of course!) to send abusive emails.

As some sins are shameful and as one cannot always get spiritual direction while in the confessional, services, such as Tor Mail, are excellent devices to get spiritual help without having to reveal one's identity, perhaps out of shame.  Most priests, I believe, will be understanding of a penitent's need to be anonymous outside of the confessional, as along as one is respectful and is acting in obedience to divine and ecclesiastical law.

I could see this being problematic. How will the priest know that he is talking to someone serious? I could see Internet trolls who hate Catholicism abusing this and wasting an already busy priests time.

I don't foresee trad priests doing this...