When one prays Minha privately on Shabbat afternoon, does he recite the verse, "Va’ani Tefilati" which is recited in the synagogue when the Torah is taken from the ark? And does he recite the Psalm of "Mizmor Shir" which is recited in the synagogue after the Torah reading?
To answer this question, we need to understand the reasons for these customs. As for "Va’ani Tefilati," the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) gives two explanations for why we recite this verse during Minha on Shabbat. The first is based on the tradition that King David noted to G-d the greatness of Am Yisrael, that on their day of rest, after feasting, they return to the synagogues and study halls to pray and to study. This is in contrast to people of other nations, who, on their day of rest, become inebriated and engage in frivolous matters. When we pray on Shabbat afternoon, that time becomes an "Et Rason" – a time when we earn special favor in Hashem’s eyes, and so we recite the verse, "Va’ani Tefilati Lecha Hashem Et Rason," which speaks of our prayers to Hashem at an "Et Rason."
The Ben Ish Hai also gives a second reason based on Kabbalistic teaching. Weekday afternoons are times of "Din" (judgment), whereas Shabbat afternoon is under the force called "Keter Elyon" ("supreme crown"), which is associated with "Rason" – G-d’s favor, and so we recite this verse which speaks of praying at an "Et Rason."
According to both these reasons, even though we recite "Va’ani Tefilati" in the synagogue when the Torah is brought from the ark, this recitation is not related to the Torah reading. Therefore, this verse is recited even if one prays privately.
The reason for reciting "Mizmor Shir" during Minha on Shabbat is discussed by Rav Shlomo Laniado of Halab (d. 1793), in his work Bet Dino Shel Shelomo. First, he writes, we recite this Psalm as a respectful way of escorting the Sefer Torah back to the ark after the reading. According to this reason, there would be no need to recite "Mizmor Shir" when praying Minha privately. However, Rav Laniado also gives a second reason, explaining that in Talmudic times, it was customary to read a section from the Ketubim as a Haftara after the Torah reading during Minha on Shabbat (just as a portion of the Nebi’im is read after the Torah during Shaharit on Shabbat). In order to commemorate this ancient practice, we recite a chapter from Tehillim (which is part of Ketubim) – "Mizmor Shir" – after the Torah reading during Minha. This reason is relevant even when one prays privately, and so one should recite "Mizmor Shir" during Minha on Shabbat even if he prays privately.
The context of Rav Laniado’s discussion is the question of why Kaddish is not recited after the Torah reading during Minha on Shabbat, as it is when the Torah is read in the morning. Rav Laniado answers that during Minha on Shabbat, the Kaddish recited after the Torah is returned to the ark, before the Amida prayer, counts as the Kaddish after the Torah reading. During Shaharit, another Kaddish is not recited until after Ashreh, and the recitation of Ashreh constitutes a "Hefsek" (interruption) in between the Torah reading and Kaddish. Rav Laniado explains that Ashreh is not really part of the prayer service, and we recite it only because of the tradition that the "Hasidim Ha’rishonim" ("early pietists"), as discussed by the Gemara, would spend some time in the synagogue after praying. We mark this practice by reciting after the service, "Ashreh Yoshebeh Betecha" ("fortunate are those who reside in Your home"), praising those who spend extra time in the synagogue. Therefore, as Ashreh is not part of the prayer service, the Kaddish recited after Ashreh cannot count as the Kaddish for the Torah reading, thus necessitating the recitation of Kaddish immediately after Torah reading. At Minha on Shabbat, however, the recitation of "Mizmor Shir" is part of the prayer service, and so the Kaddish recited afterward, before the Amida, can count as the Kaddish for the Torah reading. Hence, there is no reason to recite Kaddish immediately after the reading.
Rav Moshe Rahamim Shayo (contemporary), in his work Mehkereh Eretz, disagrees with this line of reasoning. He cites the Arizal (Rav Yishak Luria, Safed, 1534-1572) as teaching that the recitation of Ashreh is, in fact, a vitally important part of the prayer service. The word "Ashreh" relates to the Biblical word "She’er," which means "sustenance" (as in the verse, "She’erah Kesutah Ve’onatah" – Shemot 21:10). After presenting to G-d our requests in the Amida, the Arizal explained, we then receive our sustenance through the recitation of Ashreh. Rav Shayo therefore suggests that to the contrary, specifically because Ashreh is a significant, independent segment of the prayer service, it constitutes a "Hefsek," thus necessitating the recitation of Kaddish immediately after the Torah reading. By contrast, the recitation of "Mizmor Shir" after the Torah reading at Minha on Shabbat is simply a custom, and thus it is not significant enough to constitute a "Hefsek."
Summary: One who prays Minha privately on Shabbat afternoon recites "Va’ani Tefilati" and "Mizmor Shir" just as he would do if he was praying with a Minyan.