No, Obama doesn't meet the requirement like every U.S. post grandfather clause presidents, born to citizen parents.
You must be referring to the TPM constitution of the alternate universe.
Here on planet Earth in the United States Article II, Section One of the US Constitution reads:
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
And the 14th Amendment:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Obama was born in the United States, he is a natural born citizen and a citizen of the United States accordingly. He is consequently eligible to be president.
That you believe otherwise is unimportant, that Cain believes this is significant as he’s running for president, and as noted this is yet another example of Cain’s policy ignorance.
Perhaps this is the source of your confusion:
Generally speaking, a person can become a U.S. citizen in one of four ways. First, by being born in the United States or one of its territories. Second, if you were born to parents who are U.S. citizens, then you may be a U.S. citizen yourself. This process is called "acquisition" of citizenship. Third, you can be a citizen through the naturalization process, which generally involves applying for, and passing, a citizenship test. Lastly, you may be a citizen if one or both of your parents have been naturalized. This is called "derivation" of citizenship.
Citizenship and Naturalization in the U.S. Through Parents or by Birth - FindLaw
One can only assume that you mistakenly believe that acquisition of citizenship is the only way to become a citizen.
Otherwise there’s nothing in the Constitution or its case law that requires the president – or any citizen, for that matter – to realize citizenship via acquisition only.