Ibn Umar used to seek out the spot where the mushaf was placed in order to pray the supererogatory prayer there, and he mentioned that the Messenger of Allah ﷺ used to seek out that very spot. The distance between the pulpit and the qiblah wall was just enough for a sheep to pass through.
أنه كان يتحري موضع مكان المصحف يسبح فيه، وذكر أن رسول الله ﷺ كان يتحري ذلك المكان، وكان بين المنبر والقبلة قدر ممر الشاة.
Muslim narrated it in the Book of Prayer (509, 263) through several routes from Hammad bin Masada, from Yazid bin Abu Ubayd, and he mentioned it.His saying, "the pillar," means the column, and praying facing the pillar was for the purpose of a barrier, not because of any special merit attached to it.His saying, "near the mus'haf," means near the place of the chest in which the mus'haf used to be placed. Regarding the pillar mentioned, Ibn Hajar said, "Some of our teachers confirmed for us that it is the middle one in the noble Rawdah, and that it is known as the Pillar of the Migrants." Fath al-Bari (1/577). That is, the migrants used to gather beside it to talk, and this is why Umar said, "Those who pray have more right to the pillars than those who talk," which Bukhari mentioned as a suspended report. It stands today behind the mihrab, with "the Perfumed Pillar" written on it, khaluq being a type of perfume they used to apply to it.His saying, "he would pray," the word tasbih here covers voluntary prayer in general, and I have not come across any sahih hadith stating that praying beside this particular pillar carries any merit beyond the general merit of praying in Masjid Nabawi.As for the ruling on the barrier, it is not obligatory but rather an emphasized sunnah, for the Prophet ﷺ sometimes prayed without a barrier, as will come in the hadith of Ibn Abbas about the Messenger's ﷺ prayer at Mina.