Lower case 'president' is a title of an office (the office of president, in this case).
When used as a prefix before the name of the person holding the title or occupying the office, it becomes upper case: 'President'. For example, the fourth and fifth words of the article's opening sentence: 'The circumstances facing President Obama'.
Now, one can also reference that title-bearing, office-holding person using the title in the upper case: President, without using the name of the person.
I learned in high school that the definite article (the) is used when the person or object is already known to others (e.g. to readers or an audience). So I wonder why it says a President in the quoted segment when it is obvious to everybody that it is President Obama who is being referred to.
A speaker can make a definite reference to someone or something when he assumes his hearer can identify the referent. But the speaker does not have to make a definite reference in this case. So what you learned in high school was not the whole story.
Whether we say 'a President' or 'the President' depends on whether we want to make an indefinite or definite reference.
A President is an indefinite reference to this person holding this title. In addition, an indefinite reference may be specific or non-specific.
The President is a definite reference to this person holding this title. A definite reference is always specific.
At the State of the Union, a President Outgunned in Congress Is Still Combative
An indefinite reference (a) may be specific or non-specific. By the context, we know the usage 'A President Outgunned in Congress' here is a specific reference. The specific president being referred to is known from context: at State of the Union (address given Tuesday). This usage does not imply a 'lone President', but it is talking about one President, as opposed to, for example, three persons bearing the same title, as in 'The tree Tenors'.
Now, let's look at the articles in the following:
The circumstances facing President Obama as he delivered his State of the Union address Tuesday night could not have seemed less promising: a presidency with only two years left to get anything done in a Congress that is now totally in the control of a party that has routinely ignored his pleas for cooperation.
The circumstances is a definite reference. By definite, the writer assumes his readers can uniquely identify which circumstances are referred to. (What you learned in high school was similar to this, but this is a better statement regarding the the.) The reader can indeed identify which circumstances, because the whole noun phrase is the circumstances facing President Obama as he delivered his State of the Union address Tuesday night, and it is clear that the circumstances referred to are those of "President Obama."
In his State of the Union address, his is a possessive pronoun. Possessive pronouns also make a reference definite. So do the demonstrative pronouns this, that, these, those (and the archaic yon). And of course, the definite article the. These are the only classes of words in English that make a definite reference.
As for a presidency, this is an indefinite but specific reference: we know from context which specific presidency is being referred to.
And although the noun phrase "a presidency with only two years left to get anything done in a Congress that is now totally in the control of a party that has routinely ignored his pleas for cooperation" could describe a presidency other than Obama's, its use here is specific to his.