The swearing in is generally considered a formality, and under the circumstances its unlikely that anything more than a quick private swearing in would be done by the new one-hour President. The President-elect still needs to wait until noon to take office.
True Federalist
In this scenario, with the outgoing Vice President being sworn into office as President for 1 hour, then he or she becomes Acting President only? Is this correct?
That's something historians and trivia buffs would likely debate endlessly. It probably would depend on when the official word of the President's death was received.
I believe that if that was somebody else than Vice President (Speaker, PPT, Cabinet Secretary), she or he would be oficially Acting President, even if there's a vacancy and no matter for how long (an hour or an entire term). But the Vice President, in case of President's death, resignation, removal from office, automatically becomes President, with or without an oath.
Had Carl Albert succeeded Nixon in 1974, he would have been Acting President as, IIRC, no person other than VP can assume the title of the office upon a midterm succession.
Obama wasn't formally sworn-in until about 12.05 PM, but he automatically assumed the office at noon.