Wasn't it the size of the tsunami that was the problem? It was designed for a smaller scale tsunami, but that's not what happened, thus the main power backup was destroyed.
That said, if memory serves me correct, they should have ensured proper backup power a long time before the tsunami, so a timely shutdown and following cooldown could happen without problems.
It was designed for a smaller tsunami, yes, but extensive studies of the area published before construction began suggested that a tsunami of the 2011 kind was possible, and the design was inadequate. The sea wall, for example, should have been twice as tall. TEPCO ignored the studies. They also took General Electric at their word that they should install all the electrical switchgear in the reactor basements (except unit 6), despite this being the lowest point of the plant, and despite previous warnings about flooding risks (and actual flooding occurring during heavy storms predicting exactly what eventually happened). The government oversight was woefully inadequate and basically rubber-stamped whatever TEPCO said was 'acceptable'. For a country so dependent on nuclear energy, the lack of regulation and deferring to the for-profit company has led to such a backlash that all Japan's nuclear plants were shut down and remain so. By ignoring warnings, experts and everything else in the pursuit of profit, TEPCO single-handedly killed the entire Japanese nuclear industry.
That said, if memory serves me correct, they should have ensured proper backup power a long time before the tsunami, so a timely shutdown and following cooldown could happen without problems.