I was able to find a copy of the book (but only in Arabic) in the Internet Archive:

تاريخ مدينة صنعاء لأحمد بن عبد اللة بن محمد الرازي

The full title is "History of the City of Sana'a" by Ahmed bin Abdullah bin Muhammad al-Razi. He is listed as a Hafiz (one who has memorized the Qur'an) and and a Muhaddith (a scholar of the Hadith). His name would indicate he was (at least by heritage) from Ray, but I cannot find anything about him though in Rijal Al-Tusi (a book by Shaykh Al-Tusi on transmitters of Hadith, mentions a Muhaddith called Abdullah bin Muhammad al-Razi in the 10th century CE but nothing much other than that he existed. This fits as the death date listed for our al-Razi on the title page is 1068 CE/460 AH so Ahmed could be Abdullah's son though this is not provable. The introduction to the book says that he born in Sana'a, which they (the books editors) say is based on statements he makes in his own book. This is only known book so it would be difficult, based on the paucity of information about him, to say much, but the volume appears to have been well regarded in the centuries after he lived. There area even, apparently doubts as to whether he was Shia or Sunni, though the editors regard him as Sunni.

It's a long book, so the actual passage would take some digging. Enjoy!

Update: So I was avoiding real work and kept reading. The passage you are referring to starts on page 133 to 138. Like many early Islamic writers, Al-Razi freely cites multiple versions of the same story, often without comment even when these contradict. In the first version (page 133), the the Mosque is build under orders from Muhammad by Wabr bin Yuhassin. Muhammad orders him to orient the Qiblah toward Jabal Din (also called Jabal Mirmal). He repeats the story, but in the other narratives, different individuals are responsible for the mosque and for the direction of the Qiblah. The next page includes a story about the spiritual blessings of praying on Jabal Din (including that if you are in debt and pray a particular pray on the mountain, you will be released from it).

On Page 135, it tales different versions of the expansion and remodeling of the Mosque of Sana'a during the time of the Umayyad Caliph Al-Walid (705-715 CE) who ordered his governor (Ayyub bin Yahya al-Thaqafi) to undertake the work. In this story, the Qiblah is moved from its original direction to the current one (as of the author's lifetime in the 11th century CE). Who made the decision to reorient the Qiblah is given as Wahab bin Munabbih. In this version, the Qiblah prior to al-Walid's time had a deviation of "1 and 1/2 Cubits" (about 3/4 of meter or 2.5 feet) to the right (or on the right, it's a little vague). He also relates a version in which the Mosque is first built under al-Walid.

As I said, under this style of writing history, Al-Razi reports on numerous versions along with their chains of narrators. Unlike Prophetic Hadith, he does not make much of an effort to distinguish which versions are true and which not. He is reporting the traditions he has learned from men around Sana'a. Archeologically speaking, it's hard to say for certain. The oldest contemporary reference to the Mosque are from the time of al-Walid, but it's not unreasonable to assume a mosque existed before and this is the accepted account. There have been extensive archeological digs at the Mosque leading to many important documentary finds including the Sana'a Palimpsest.