What should I do to resolve an incorrect birth date on my FAA medical certificate?
I read the FARs for this one, and there is a rule:
14 CFR §67.403 Applications, certificates, logbooks, reports, and records: Falsification, reproduction, or alteration; incorrect statements.
From what I gather, the rule is there to get liars in trouble.
If you didn't lie, which you clearly have no reason to because your age does not disqualify you, I don't think you have to worry about getting in trouble because it was the doctor's mistake.
However: (From 67.403)

(c) The following may serve as a basis for suspending or revoking a
medical certificate; withdrawing an Authorization or SODA; or denying
an application for a medical certificate or request for an
authorization or SODA:
(2) An incorrect entry, upon which the FAA relied, made in any
logbook, record, or report that is kept, made, or used to show
compliance with any requirement for a medical certificate or an
Authorization or SODA. [My emphasis]

Let's pull out the important words and we get this:
&quot;An incorrect entry ... made in any ... record, or report that is kept ... or used to show compliance with any requirement for a medical certificate...&quot;
A medical certificate is certainly kept or used to show compliance with a requirement to posses a medical certificate §61.3(c)(1)
You have discovered an incorrect entry.  Call the doctor that issued your medical as soon as you can in order to get it resolved, and do not accept a solo sign-off on this incorrectly filed document now that you have knowledge of the mistake.
To give you a straight answer based on my interpretation of the law, yes; this error would prohibit you from soloing until it can be corrected.