From the Safer Haminhagim: [Every day from Rosh Chodesh Nissan until the twelfth of the month, usually after Shacharis,] one reads the passage [from Bamidbar 7-8:4] that describes the offering brought on that day by a particular Nasi, or tribal prince, for the dedication of the altar of the Mishkan. [In common parlance, each day’s passage itself is often referred to as “the Nasi.”] This daily reading is followed by the prayer which opens with the words yehi ratzon (and which appears in Siddur Torah Or [as well as in Siddur Tehillat HaShem, p. 371]). This prayer is recited even by a Kohen or a Levi [despite its seeming relevance only to tribes other than the Tribe of Levi]. 
Footnotes:  HaYom Yom, p. 41. (According to the instructions given for this reading in Siddur Torah Or and Tehillat HaShem, on the thirteenth of Nissan one should read [the summarizing paragraph which begins] zos chanukas mizbeiach (i.e., from Bamidbar 7:84 to 8:4). This would appear to be an oversight, because in his Shulchan Aruch (429:15) the Alter Rebbe writes that on the thirteenth day one should read from [the beginning of] Parshas Behaalos’cha until kein asah es hamenorah (i.e., Bamidbar 8:1-4), a passage that corresponds to the Tribe of Levi. – The above comment is among the last of the notes with which R. Avraham Chayim Naeh introduces his Piskei HaSiddur.) The actual practice to be followed requires clarification.