The temple dates back to 200 BC, although the present structure probably dates back to the 9th century AD. It was visited by Adi Shankara and has ever since been associated with him; this is how the temple got the name Shankaracharya. It is also regarded as sacred by Buddhists. The Shiv ling was placed inside during the Sikh period in nineteenth century and it became an active Hindu temple when regular services were conducted. Some historians report that the temple was actually a buddhist temple during buddhist era which was then changed into Hindu site of worship by Adi sankaracharya. Persians and Jews call it Bagh-i- sulaiman or the Garden of King Solomon. Persian inscriptions are also found inside the temple.
According to Pandit Anand Koul (1924) the temple was originally built by “Sandiman” (unknown) who, according to Koul, reigned in Kashmir from 2629 to 2564 BC. It was repaired by King Gopaditya (426–365 BC) and by King Lalitaditya (697–734). Its roof was also repaired by Zain-ul-Abidin after an earthquake; later, its dome was repaired by Sheikh Ghulam Mohiuddin, a Sikh governor (1841–1846).
The earliest historical reference to the hill comes from Kalhana. He called the mountain Gopadri. Kalhana also says that King Gopaditya granted the land at the foot of the hill to the Brahmins that had come from the “Araya versh.” The land grant was called “GOPA AGRAHARAS”. This area is now called Gupkar. Kalhana also mentions another village in the vicinity of the hill. It so happens that the King Gopaditya housed some of the Brahmins who had eaten Garlic to a village next door. Kalhana names this village as Bhuksiravatika. That would be Buch’vore today. Kalhana also mentions that King Gopaditya built the temple on the top of the hill as a shrine to Jyesthesvara (Shiva Jyestharuda) around 371 BC. Abul Fazal also mentions that King Gopaditya built the temple. Although many experts believe that the current temple was probably built later but most of them agree that the base of the structure does seem to be very old. It is said that Lalitaditya Muktapida (724-726 AD) of Karakote dynasty did repairs to the temple. According to Srivara Budsah (Zain-ul-Abideen 1420-1470 AD) did major repairs to the temple. He also put the Kalash (spire) and the roof of the structure which had fallen due to an earthquake.The third time the temple was repaired was by the Muslim Governor, Sheikh Mohi-ud-Din (1842 to 1845) during the Sikh rule of Kashmir. The name of the hill was changed from Gopadri to Takhate-Suleiman some time during the period of Sultans. We have no idea why the hill was given that name, as there is no historical evidence whatsoever that King Solomon would have ever come close to our valley. The fact that the mountains in the North West of Pakistan are also called Kohi-Suleiman seems to point to the fact that many mountains were named after Suleiman by Muslim kings. The name Takhate-Suleiman however seems to have persisted during the Mughal, Afghan, Sikh and Dogra periods. In fact if you look at any publication during these periods they all refer to the hill by that name. The name Shankaracharaya for the temple on the hill (not the hill mind you) first appears when Governor Sheikh Mohi-ud-Din made the repairs in the mid-19th century. It had apparently not been used as a temple for centuries. At that time the temple was consecrated as a Shankaracharaya temple and a Shiv Lingam was placed in the temple. It is only during the Sikh period that people started to have prayer services at the temple and the Shravan Poornima started being celebrated at the temple. The current theory that Adi Shankaracharaya came and meditated at this temple is not bourne by any historical sources. Shankaracharaya would have done so in the 8th century and Kalhana would have definitely mentioned it. So the name Shankaracharaya for the hill has as much historical validity as Takhate-Suleiman. The Dogra King Gulab Singh (1846-1857 AD) constructed the steps to the hill from Durga Naag temple side. The Maharaja of Mysore came to Kashmir in 1925 and he made the electrical installtions at the temple. In 1961 Shankaracharaya of Dwarkapeetham put the statue of Adi Shankaracharaya in the temple. In 1974 the Government of J&K constructed the road that goes to the TV antenna near the top of the hill.
(source:wikipedia). The address of Shankaracharya Temple, Srinagar is Srinagar, Jammu and Kashmir
It's in Srinagar, Jammu and Kashmir