Why does the Torah not mention "Nasi" regarding Nachshon ?
Da'as Zekeinim mi'Ba'alei Tosfos: To prevent Nachshon from becoming conceited, for being the first to bring his Korban. 1
Gur Aryeh: It is to teach that he did not offer first due to his own importance, rather, due to his Shevet.
Ohr ha'Chayim: This teaches that he was proper to offer first due to his own level - he was the first to jump into the Yam Suf - as opposed to the others, whose only attribute was being Nasi of a Shevet.
Ha'Emek Davar #1:We find that the greatest Tana'im 2 were not called Raban. Similarly, the greatest Nasi was not called Nasi.
Ha'Emek Davar #2: Each other showed his Nesi'us and authority on his day of Simchah. Nachshon had only Simchah of Mitzvah on his day, for he was sad due to the death of his sister's sons (Nadav and Avihu).
Moshav Zekeinim citing the Rosh: Shevet Yehudah learned to belittle themselves from Yehudah - "Yeshev Na Avdecha Tachas ha'Na'ar Eved" (Bereishis 44:33). And it says "v'David Hu ha'Katan" (Shmuel 1:17:14).
I.e. the Zugos. (We find that he was called Nasi above (2:3). Below (34:19-28), regarding the Nesi'im for the division of Eretz Yisrael, it says Nasi for all except for the first three (Yehudah, Shimon, Binyamin). Emes l'Yakov there gives a reason. (PF)
What is the significance of the 'Hey' in ba'Yom ha'Rishon"?
Rashi: Refer to Vayikra 9:1:3:1*.
Why does it add "le'Mateh Yehudah"?
Rashi: It is in order to attribute him to his tribe. 1
Refer to 7:12:1:2.
Rashi: But not because he claimed the money for his Korbanos from the members of his tribe - since the paragraph ends with the words "Zeh Korban Nachshon ben Aminadav", indicating that he paid out of his own pocket.
Why does it say "va'Yhi ha'Makriv ba'Yom ha'Rishon"?
Da'as Zekeinim, Rosh: It was not proper that he offer first, for Reuven and Shimon are older, just Yehudah is always the first. Refer to 2:3:1:1*.
Hadar Zekeinim: Yehudah merited to be first, because Binyamin wanted to enter the Yam Suf first, and Yehudah stoned them and they entered first. 1
Rosh #2: Nachshon offered first, for he was the Gis of Aharon (their wives were sisters).
Perhaps we can read this like 'ba'Yam Rishon', even though ba'Yom has a Vov, just like Megilah 25b says to read "Kevodo" like 'Keveido', even though Kevodo is written with a Vov! (PF)