Genetic Diversity and Adaptation
Great question! A lot of things affect how quickly a population or species can adapt to a new environment, including population size, mutation rate, generation time, standing genetic diversity, and selective pressure.

The diversity of life encompasses practically all combinations of those variables. A bacterial population might very well contain enough diversity to allow a portion of the population to overcome a rapid change. In fact, applying antibiotics to a bacterial population and counting the survivors is a common measure of mutation rates.

On the other hand, organisms with small populations and long generation times will be much less likely to overcome a rapid environmental change. This is why there is so much concern over anthropogenic changes to the environment, including climate change.

It's hard to provide a definitive answer, since "rapid change" is a relative term, and the difficulty of adapting isn't known. Some striking adaptations can be caused by a single base pair mutation, such as in beetles and Monarch butterflies that are insensitive to toxic plant compounds.

It's important to point out that natural selection is based on relative fitness. Therefore, an adaptive mutation will spread because it's likely that carriers will be more fit than all of their neighbors. This does not necessarily mean that individuals without the mutation will die or fail to reproduce, only that those with the mutation will do it better.

Likewise, adaptation is not necessarily caused by the environment changing and wiping out all but a few lucky mutants like in the bacteria example. Instead, the environmental change might make things more difficult, but as long as the population can persist, then mutations will continue to enter the population which could confer a selective advantage against the change. So, no, a population does not (cannot) carry all possible adaptations. A population cannot adapt to an environment it hasn't encountered.

Finally, it's worth pointing out that a species can expand its range through migration. A new, unsuitable environment can act as a migration sink (that is, migrants make it there but fail to establish) for a nearby population. If this happens for long enough, some of the migrants may have a mutation allowing them to establish in the new environment.