Estimate moonrise and moonset times.
At new moon the moon rises with the sun, and each day thereafter it rises roughly fifty minutes later — the lag between the lunar day and the solar day. Adding that lag multiplied by the moon's age to local sunrise gives a workable moonrise estimate, and the moon stays above the horizon for about twelve hours and twenty-five minutes on average. The lag varies with latitude and season, which is why it is an input rather than a constant.
Moonrise from lunar age
Moonrise = local sunrise + moon age x daily lag; moonset = moonrise + 12 h 25 min above the horizon
The moon moves about 13 degrees eastward along its orbit daily, so the Earth has to rotate a little further to bring it into view — around fifty minutes on average.
Because it varies from about 20 to 70 minutes depending on latitude and the moon's position along the ecliptic. The harvest moon effect is a well-known case of an unusually short lag.
This is a planning approximation, typically within tens of minutes. For precise times use an ephemeris that accounts for your exact coordinates and the moon's declination.