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.