Rice is often replaced with noodles or bread with dishes of Western origin, and soup is often omitted when the meal contains another soupy dish or at dinner when having drinks.
At most Japanese tables, everything comes at the same time* and dishes are eaten in alternate order.
*At many Japanese restaurants in the US, miso soup often comes first -- probably in an attempt to conform to the serving order of Western food -- but it’s extremely odd to have a bowl of miso soup without any other dishes, especially rice. If you ask that the miso soup be served with your meal, you will get special respect as someone who understands Japanese culture ... maybe.When the meal is nabe hot pot, which is usually large and shared with a number of people, rice or noodles (usually udon noodles) are eaten at the very end by adding them to the soup after other ingredients in the pot are finished.
With formal style meals at pricey restaurants, dishes are served in a fixed order, just like a course meal. Also, when drinking is part of the scene at a restaurant, the rice, usually a small amount, comes at the end -- it is like sealing your stomach for the night.
What goes where on your table