You could slowly evaporate water, forming clouds, which blow with the wind (the same direction as the clouds in the game, whatever that is). The cloud could release its rain proportionately to high how the ground is below it, so that for example when the cloud approaches a mountain that extends to the max height, it releases all of its water on the mountainside, like in real life.

This way, you'll also have "rain shadows" on the side of mountains facing away from the direction of the prevailing winds.

You'll end up with naturally forming streams, lakes, rivers. Some lakes will dry up completely. Eventually you'll reach a steady state equilibrium like we have in real life, with more or less stable water levels, rivers, ocean levels, etc.