We report the measurement of gamma gamma -> eta(c) (1S) -> eta(c) (2S) -> eta'pi(+)pi(-) with eta' decays to gamma rho and eta pi(+)pi(-) using 941 fb(-1) of data collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider. The eta(c) (1S) mass and width are measured to be M = [2984.6 +/- 0.7 (sta) +/- 2.2 (syst) +/- 0.3 (model)] MeV/c(2) and Gamma = [30.8(-2.2)(+2.3) (stat) +/- 2.5 (syst) +/- 1.4 (model)] MeV, respectively. First observation of eta(c) (2S) -> eta'pi(+)pi(-) with a significance of 5.5 sigma including systematic error is obtained, and the eta(c) (2S)mass is measured to be M = [3635.1 +/- 3.7 (stat) +/- 2.9 (syst) +/- 0.4 (model)] MeV/c(2). The products of the two-photon decay width and branching fraction (B) of decays to eta'pi(+)pi(-) are determined to be Gamma(gamma gamma)Gamma B-gamma gamma = [65.4 +/- 2.6 (stat) +/- 7.8 (syst)] eV for eta(c)(1S) and [5.6(-1.1)(+1.2) (stat) +/- 1.1 (syst)] eV for eta(c) (2S). The cross sections for gamma gamma -> eta'pi(+)pi(-) and eta'f(2)(1270) are measured for the first time.