The Moon is highest about one hour after the Sun for every 15° it is east of the Sun: a new moon at noon, a first quarter at 6 pm, a full moon at midnight and a last quarter at 6 am.
The sky turns 15° an hour, and a phase is the angle between the Moon and the Sun: 90° at first quarter, 180° at full. A Moon 90° east of the Sun crosses the meridian 90 ÷ 15 = 6 hours after the Sun does, which is about 6 pm, and one 180° away crosses it 12 hours after, at midnight. The same rule says when each phase rises and sets: a first quarter is up from noon until midnight.
It is solar time, not the clock, so add the difference between solar noon and 12:00 (up to an hour or more in summer time). The Moon moves 12 to 15° a day against the Sun, so the angle used is the one at the time you look, and the height at which it crosses depends on the season and on where the Moon is in its tilted orbit: a full moon is high in winter and low in summer, the opposite of the Sun.
Estimate with the rule, then check it against the calculator that models it properly.
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