Justin: The Spanier-Whitehead dual of K(C_p,2) is a point! This was proved by Chun-Nip Lee (a student of Gunnar Carlsson, now not in academic math): see Thm 2.5 in On Stable Maps From Eilenberg-Maclane Spaces to Classifying Spaces
Chun-Nip Lee American Journal of Mathematics Vol. 114, No. 2 (Apr., 1992), pp. 405-412. One analyzes things the way Tyler suggests, and the calculations mirror the calculations that these same Eilenberg-MacLane spaces are acyclic in K-theory.