I feel pretty confident that clocks and calendars are going to continue moving forward at more or less the same rate they have been since time became the abstraction it is. The Earth will continue orbiting the sun at the same speed, while keeping on its same steady rotation. Microwave frequencies, the measuring stick used by atomic clocks, will keep on as they always have. The world might end in an endless number of ways but, in most of them, time keeps marching forward.
Unless it doesn't. What if instead of a bang or whimper, the universe simply grinds to a halt? It's a real if highly speculative idea. It might not even be that much more unlikely than all of the other end-times scenarios you've heard about: grey goo, asteroid collision, AI overthrow. Better hope you're not doing anything embarassing when the Big Stop goes down.
See, we have a neat set of physical laws. Setting aside our endless battle for a Grand Unified Theory, they're still laws that should describe everything. But there's a problem with that in the endless Universe. In an endless Universe, everything necessarily happens. Duplicate Earths, for example. When everything happens an infinite number of times, as would have to happen in an endless Universe, then it becomes impossible to apply probabilities to anything.
Without probability, we lose the physics that enable something like time to exist. A group of doom-nerds (Vladimir Rosenhausa, et. al.) at the University of California, Berkeley described this in a 2010 paper: Time will cease to exist at a certain point. Specifically, they have it that there's a 50 percent chance that time will end within the next 3.7 billion years. We don't know how it will happen and, likely, we are just plain unable to know how it will happen.