Before the release of Gathering Storm, this is a largely useless improvement, even for a Religious Victory player. A yield of 2 Faith per improvement is negligible, and the idea of retreating your religious units all the way back home to heal up is impractical; especially when considering the Religious Victory has the most narrow opportunity window of all. On a standard sized map or above, no one should ever spend turns to march their religious units to the civilization half across the map, get them damaged, march them back home to heal and mobilize them again. Armagh's suzerainty, therefore, should be (almost always) dead last on your priority list.
After the release of Gathering Storm, however, the Monastery receives a buff that makes it actually useful, it provides 1 Housing per improvement, 2 Housing with Colonialism with additional Faith based on adjacent districts. Remember, 1 Housing early game is worth 2 Farms, Plantations, Pastures and Camps, and this improvement can be placed anywhere, just not next to each other, unlike the Stepwell, the Kampung or the Mekewap. This improvement alone can pretty much single-handedly solve every Housing issues that a civilization can run into, thus civilizations with Food bonus like India, Khmer or the Incas should definitely compete for this City-state.
Of note: The Gathering Storm bonus is not currently active on the Switch version of the game
A monastery, in the Roman Catholic tradition, is a complex of buildings that serves as communal home and workplace for monks (and nuns, in modern usage termed “convents”). These also include someplace reserved for prayer, such as a chapel or church. And many housed hospitals, schools, distilleries, and more. According to legend, Christian monasticism began in Egypt, although the first monastery in Europe was founded c. 344 AD near Chirpan (Bulgaria) by St. Athanasius. A couple centuries later, in 529, Saint Benedict established the monastery at Monte Cassino, laying the pattern for Catholic monasticism. Monasteries soon began appearing all over Europe, and beyond.