Bearing in mind that Nachshon brought his Korban on Sunday (See Pasuk 13), how could Elishama ben Amihud bring a Korban Yachid on Shabbos?
R. Bachye (citing Midrash Rabah): It was a Hora'as Sha'ah (a temporary ruling). 1
Da'as Zekenim: Because Yosef kept Shabbos before the Torah was given, as the Torah indicates in Vayeishev, Bereishis 43:16 "u'Tevo'ach Tevach Vehachein. 2 "
R. Bachye (citing the Tanchuma: Because Yosef kept the Mitzvh of "Lo Tin'af" (the seventh commandment - R. Chavel's footnotes) before the Torah was given, his children merrited being the only ones who were ever permitted to bring a Korban Yachid on Shabbos. See R. Bachye.
Moshav Zekenim: Since "Vehachein" hints at Shabbos - as in Beshalach, Sh'mos, 16:5 "ve'Hayah ba'Yom ha'Shishi Veheichinu ... ". Also the Pasuk in Tehilim 60:9 "Li Gil'ad... ve'Efrayim Ma'oz Roshi ..." alludes to [Hora'os Sha'ah - a. to the Korban of] Eliyahu on Har ha'Karmel, of Gidon, and b. to the Korban of the Nasi of Efrayim on Shabbos; Also Yehoshua who was from Efrayim conquered Yericho on Shabbos, (PF). See also R. Bachye p. 40 - end of Dibur.


