The non trivial zeros respective poles of $\zeta$ lie outside $ Re s > 1$ follows from the absolute convergence of the Euler product in this region and does not need any entire function theory, not even holomorphicity. That the Riemann Zeta function is holomorphic, actually follows from the absoulute locally uniform convergence of the product. The functional equation follows from the Poisson summation formula. Having the functional equation and the nonvanishing established, we could apply a contour integration, but we need certain boundedness conditions in the critical region, which needs the Phragmen Lindeloeff principle (not entire function theory, but this is already a compex analysis argument), then the explicit formula follows.
If you interested in the merely existence of nontrivial zeros, you might want to try to deduce this then from knowledge over the primes by inserting an appropiete chosen test function here. It might be helpful to use the analogy with the Selberg trace formula and the derivation of the Weyl law. Werner Mueller has some nice expository articles here.
One proof using only the languages of the adeles, harmonic analysis and no entire function theory explicitely at all was given by Ralf Meyer: http://arxiv.org/pdf/math/0311468v3
However the Fourier transform of compact supported function is always entire, so you basically just hide the complex analysis arguments.