How often does the moon "set" for us compared to the sun?
As noted in comments, the earth rotates on its axis in about 23hrs 56 minutes. This is known as the sidereal day, that is, the day with respect to background stars. (that's not quite true, but it's good enough unless you're dealing with periods of thousands of years with high accuracy - see the linked wiki article if you care about the detail).

At the same time, the Earth is also orbiting around the Sun in the same direction, and the combination of the Earth's spin and its orbit around the Sun gives us the 24-hour solar day that we are familiar with, and that you have already modelled in your game.

Now consider how the Moon rises and sets, when seen from a position on Earth. The primary cause of this is, again, the Earth's rotation with it's just-less-than-24-hours period. In addition, this time, one must consider the motion of the Moon around the Earth, which is again in the same direction as the Earth's spin. This results in a period of about 24 hr 50 mins for the apparent motion of the Moon across the sky (Source). For your game that may be sufficient info, although if you want to be accurate it is complicated.

Remember to also take account of the phases of the moon - because the whole earth-moon system is also in orbit around the sun, and because the moon is only visible via reflected light, the mere fact that the moon is above the horizon when seen from a given point on earth does not mean that it is visible.