The entire world is already supplied with underground shelters for the apocalypse of various sizes, from the basements of urban buildings prepared as civil defense shelters in the 1950's to underground command and control bunkers in the United States, Russia, China, the UK, France and in many other nations as well.
The vast build out of these shelters during the Cold War was in principle to shelter sufficient people and equipment to allow for rebuilding after a nuclear war. Some were fairly rudimentary, for shelter during the immediate attack, like civil defense shelters in the US and the Moscow subway system. Others were fully equipped long term systems such as the NORAD defense complexes in Colorado and North Bay. There are also alleged "secret" bunkers, such as the complex supposedly dug under the Denver airport.
There are several ways this could be handled in story terms. Fully equipped and functioning bunkers already exist, but are owned and manned by military of government personnel. These people are also the best organized and equipped to deal with long term existence underground and the ability to recolonize the surface afterwards.
Many cold war bunkers were decommissioned after the Cold War ended, but since they were built to withstand nuclear attack it is impractical to dismantle them. It may be possible for civilians to have access to them, such as the "Deifenbunker" near Ottawa, Ontario, Canada, which has been restored as a museum. If the inductions that war was coming were clear and with enough advanced notice, the museum staff could possibly stock the bunker and seal the doors. It would not work as well as the fully stocked and prepared bunker in the 1950's and 60's, but would be far better than anything else. There are also people who have purchased decommissioned nuclear missile silos who would have the same intrinsic level of protection.
It is very unlikely that even the NORAD bunker in Colorado Springs would be able to be viable if sealed for a century, but then again, even an all out nuclear war would not be as damaging as you seem to think. Nuclear weapons are much smaller in power than during the height of the Cold War (yields in kilotons rather than megatons), and nuclear stockpiles have been drawn down over the years from the tens of thousands in the Cold War era as well. The heightened accuracy of delivery systems means fewer and smaller weapons are needed today.
So long as the bunkers can be sealed prior to the attack, they can survive for periods ranging from days to years, but that is all they might need. So there is no requirement for an Elon Musk to build a secret underground city: there are already hundreds of bunkers around the world built for that purpose already.