Dual of a holomorphic vector bundle
Solution 1:
I am contemplating about the same thing right now. The cocycle construction of dual bundle is easy for definition but it does not lie in my head solidly. The following link seems to have some clues about how to construct it in terms taking duals of fibers of a given vector bundle. I will update my answer if I figure out something more.
http://www.math.sunysb.edu/~azinger/mat531-spr10/vectorbundles.pdf
Update: I have edited my answer in great detail, but it somehow was rejected by the website. I believe the answer to your question is just a matter of which definition you take, and whichever definition you take, the holomorphy of $E^{*}$ is given by that of $E$ because the charts of $E^{*}$ is induced by local biholomorphic maps $E(U) \simeq E^{*}(U)$ given in the data of local trivialiaztion. $E^{*}$ is defined to be holomorphic, which is funny that one should prove it, if I understood correctly.