You mean the one buried under the ice caps of Antarctica?
Given that we're still discovering the remains of ancient cities thanks to satellite photos, I'd say the answer to your first question is "fairly big".
To try to narrow down the size a bit, I refer to this article about ancient cities in the Amazon. Many civilizations existed and have been identified in the Amazon floodplains, but the less fertile uplands where thought to be empty. Recently discovered remains hint at a population size of 60,000 people, rather than 0.
It took us until this century to find (part of) a civilization of 60k people in a remote jungle-like location. I'd say that is a good starting point. You might get away with a greater size if the location is more remote or covered by plants/mud/ice/etc. The closer the site is to our current civilization (especially construction work and agriculture), the quicker it will be discovered, so that makes your possible population size much smaller.
The options for city-wiping disasters are fairly limited, so let me try to list them:
- Volcanic eruption covering the city in lava and ash. Very effective in burying evidence, but it would have to be huge to cover multiple cities.
- Catastrophic flood. Even without divine wrath these can ruin your day. Civilizations located entirely within one valley would be vulnerable to this. An avalanche at the valley's mouth might turn the entire valley into a lake, forcing the inhabitants to abandon all their advanced construction. A natural (or man-made!) dam collapsing at the head of the valley might instantly bury the civilization in mud.
Note that a tsunami definitely could wipe out a coastal civilization, but the remains would likely be discovered easily, because humans have followed coastlines while exploring since the first ones decided they "needed some space to think".
- Ice age. The advancing ice would have forced the civilization to go nomadic or extinct. If they were located on Antartica, they might have gotten trapped there or died on the journey to other lands. A handful might have reached Australia, South America or various islands.
Finally, the civilization might existed in a jungle or on floodplains and have died out because of disease or famine, after which nature would have covered it up fairly quickly by either one of the disasters above or gradual processes such as seasonal flooding or plant growth.