From the perspective of someone falling in, the outside universe experiences an infinite amount of time. So if it's going to end, it's going to end before the infalling observer has even the very short period of time required to cross the event horizon.
If the universe doesn't end, it will have infinite time to cool off and the black to hole to evaporate from Hawking radiation. To conserve energy that means the infalling observer must observe a greater and greater amount of Hawking radiation the closer he gets to the event horizon, and the horizon will always recede from him as the hole shrinks. When he eventually reaches the centre he'll discover that there's no black hole left.