It isn't even quite that clear: Whether he is eligible to run as a VP candidate is a Constitutional question that has been discussed by legal scholars ad infinitum. For starters, the 22nd Amendment to the Constitution says:
"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once."
That seems to be loophole; he would not be elected to the office of President, but to the office of Vice President. So far, so good - and as you pointed out, the 12th Amendment reads:
“...no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”
The real legal issue - and it would have to be eventually sorted-out by the Courts - is whether "...constitutionally ineligible to the office of the President.." means ineligible to run for that office, or ineligible to serve in that office.
My specialty is not Constitutional Law, but my reading says that Obama could very well be eligible to run. Knowing this would, sooner or later, wind up all the way in the Supreme Court, and knowing the extremely political make-up of that court today, I would have to say.... all bets are off.
At this point - unless the "birthers" are going to contend that Obama is actually a woman in disguise - it is academic anyway. It is, however, and interesting topic for debate - and that debate probably will never be settled until someone actually tries to make the run.
Using logic, if you asked people what are the requirements that make eligible to run for president, they'll usually point to the ones described in Article II. But again, the 22nd Amendment adds an additional layer to the eligibility criteria by restricting two term presidents from the office ever again. It says nothing about the VP. The question is whether the Amendments are meant to be independent of each other unless explicitly addressed (see 18th and 21st Amendments) or whether they only apply in the order in which they come.
Think of this as a lease agreement. You have a base lease agreement that stipulates the terms of the lease and you also have some addenda which expand on details from the base agreement. If after a few months of the agreement being in place another addendum needs to be added, it will almost always tell you what it's amending or expanding while making a reference that everything else remains intact. The 22nd Amendment failed to do this since it says no person shall be ELECTED to the office of president. The word ELECTED is the problem here. Had they used another word, such as sworn or accede, it will have likely carried bigger weight. Obama could have run for a 3rd term and been elected, but the 22nd Amendment would bar him from taking the oath of office and serve his 3rd term (therefore, he wouldn't run). Likewise, if he runs for VP, he could serve the term but he'd have to pass up the oath if the president was unable to carry out the duties. In such a case, parties would not even bother because they would risk the presidency falling to the Speaker, which could be from another party. Additionally, this also safeguards against the potential scenario where a former president is nominated to lead a department and the line of succession demands they take the oath. Imagine being nominated to SoS, survive an attempt against the government but the president, VP, Speaker, and PPT of the Senate are all dead; as the 4th in line, Obama would be sworn in.
Of course, if the intent is to discourage a president from running to the executive office again (as a VP) and any chance of them getting close to the Oval Office in any other capacity, then the amendment could have had a third section (which would be Section 2, pushing the existing Section 2 down to Section 3) which states that a person that meets the criteria of Section 1 is also ineligible to SERVE as vice-president or accede to the presidency through the line of succession (effectively closing the loophole by not allowing them to be elected to either the presidency or vice-presidency, be appointed as vice-president, or be sworn in as president in the event that they're high in the line of succession and it's their turn).