1)
Who is this Nasi?
1.
Rashi: He is the Kohen Gadol. The same applies to every Nasi in this discussion.
2.
Rashi citing Menachem: He is the king.
3.
Radak, Malbim (44:3): He is Melech ha'Mashi'ach ? "v'David Avdi Nasi Lahem l'Olam" (37:25).
2)
Does the Nasi offer for them only during the Mo'adim?
1.
Malbim: No ? he offers Olos, Minchah and Nesachim the entire year. Olos Temidim are brought from his money. On the festivals, Rosh Chodesh and Shabbos they bring additional Korbanos for Musaf, Chatas goats and lambs on Shavu'os for Shalmei Tzibur. He pays, and they are Yotzei, for all joined with him (gave to him Terumah).