Ani is a ruined medieval Armenian city now situated in Turkey's province of Kars, next to the closed border with Armenia. According to Wikip[edia it was indeed known as the City of a Thousand and One Churches.

Established as the capital of Armenia, in 962, it gradually declined over centuries until the site was entirely abandoned by 1735—when the last monks left the monastery in the Virgin's Fortress or Kizkale.

Clearly from the picture, there is very little standing in Ani, and it has probably been that way for centuries.

The city would nowadays be called Vagarshapat. It is an ancient city. Gurdjieff provides Vagarshapat as alternative name to Echmiadzin.

The city is best known as the location of Etchmiadzin Cathedral and Mother See of Holy Etchmiadzin, the center of the Armenian Apostolic Church. (It is here where the Catholicos of all Armenians resides and carries out his religious duties.) It is thus unofficially known in Western sources as a "holy city" and in Armenia as the country's "spiritual capital". It is one​ of the most common pilgrimage destinations of Armenia.

We have been unable to discover any reference to a "Kanzaran"—a place where treasures are kept—in Echmiadzin in relation to the Armenian church. lf such a place by that name exists it is not publicized.

The word Kanzaran may be of Hebrew derivation. with Kan meaning "here" and Zaran meaning "God has richly blessed." This implies a place where things that God has richly blessed are stored.

There is no evidence on Google Maps of a village close to Alexandropol (now called Gyumri) with the name Karadagh. The name is Turkish and means Black Mountain. It is possible that the village was renamed or merged into Gyumri.

The area known as Arasbaran which stretches from the Qusha Dagh massif, south of Ahar, to the Aras River in East Azerbaijan Province of Iran is also known as Karadagh. It seems unlikely that Gurdjieff is referring to that in the text, but it is worth noting.

Karklis is now known as Vanadzor. Karakilisa means "black church" in Turkic. It is the third-largest city in Armenia and the capital of Lori Province, about 128 kilometres (80 miles) north of the capital Yerevan. Vanadzor is the seat of the Diocese of Gougark of the Armenian Apostolic Church.

The Moushtaïd Garden in Tiflis has been called the "Bois de Boulogne" of Tiflis. It was established between the 1830s and 1840s, by Mir-Fatah-Agha, a high-ranked Shia Muslim religious leader of the Persian Shiites. He was a Mujtahid and the garden was named for that. In the Shia tradition, a Mujtahid is a person who is accepted as an original authority in Islamic law.

In 1858, the garden became a public recreation centre.

Pera (also known as Beyoglu) is a district of Istanbul on the west side of the Bosphorus. Pera is the Greek name for the district.

Wikipedia link

It may be that the Armenian church of Sev-Jiam, Gurdjieff refers to, where there is a miraculous icon of the Virgin is the church of St Astvatsatsin or the Yot Vera (Seven Wounds) Church. In Armenian sev-jiam means "black horse."

The Tiflis Soldiers’ Bazaar is most likely what is now called the Dezerter Bazaar. It acquired that name around 1920 because many deserters from the First World War sold their weapons there. It is now, primarily, a fruit market.

The Solomon Islands are East of the Sunda Islands. Solomon, of course, signifies wisdom, more than any other figure from the Bible.

The name Sunda derives from the Sanskrit prefix su, which means "goodness" or "possessing good quality". The term sunda means "bright, light, purity, cleanness and white." Sunda is also another name for Hindu God Vishnu the preserver.

Vagharshapat  is the 4th-largest city in Armenia. It is commonly known as Echmiadzin, which is the name that Gurdjieff prefers.

The area of Vagharshapat was once known as Artimed—from the ancient Greek goddess Artemis. Later, it was renamed Avan Vardgesi (Town of Vardges) or Vardgesavan by Prince Vardges Manouk who rebuilt the settlement near the shores of Kasagh River, during the reign of King Orontes I Sakavakyats of Armenia (570-560 BC). The Byzantine historian Procopius refers to the city as Valashabad (Balashabad), named after king Vologases I of Armenia. The town of Vardges was entirely rebuilt and fenced by King Vagharsh I to become known as Norakaghak and later Vagharshapat.