What's the size of the plastic cup that came with my rice cooker?
Assuming you bought a rice cooker designed by a Japanese company (and apparently even other brands tend to meet that market's expectations), the measurement is 1-gou, slightly more than 180ml, which, by no coincidence, is also the typical measure of a wooden sake, cup, and is closely associated with a historical sake bottle size (approximately 1.8l)

http://en.wikipedia.org/wiki/Masu_(Japanese)

It turns out that this amount, 1 gou of dried rice, neatly corresponded to a typical serving of rice. In practice, most contemporary Japanese eat about 1.5-2 gou per day; 1 gou of dried rice cooks up enough for 2 Japanese adults for one meal if you have several side dishes. There are other measurements that derive from the gou (or perhaps the other way around), such as the koku, which was considered the amount of rice that a single person would consume over the course of a year.

This is one of the human-centric forms of measurement that has survived the metric push; you can find various examples of this in many otherwise metric-converted countries. It turns out some studies show that those metrics often make certain categories of estimation easier for people.

Edited to cover the concern about matching the right amount of water:

It's worth noting that you don't need perfect precision for the amount of water, as long as you cook with the full cycle and not one of the express cooking modes. I can't remember the exact scientific principle behind it, but perhaps something to do with osmotic pressure. Some people use the remarkably effective method of measuring a certain amount of space between the dried rice and the water based on the size of their forefinger segment or thumbnail. It apparently works well for almost any imaginable size of pan (though you can have other problems with a pan too wide to have the rice cover the bottom). (Some types of rice do prefer more water than others, but within a single type, you have a fairly flexible range for the water ratio)