We consider the phenomenology of a class of gauge-mediated supersymmetry (SUSY) breaking (GMSB) models at a e+e- Linear Collider (LC) with c.o.m. energy up to 500 GeV. In particular, we refer to a high-luminosity (L ~ 3 x 10^34 cm^-2 s^-1) machine, and use detailed simulation tools for a proposed detector. Among the GMSB-model building options, we define a simple framework and outline its predictions at the LC, under the assumption that no SUSY signal is detected at LEP or Tevatron. Our focus is on the case where a neutralino (N1) is the next-to-lightest SUSY particle (NLSP), for which we determine the relevant regions of the GMSB parameter space. Many observables are calculated and discussed, including production cross sections, NLSP decay widths, branching ratios and distributions, for dominant and rare channels. We sketch how to extract the messenger and electroweak scale model parameters from a spectrum measured via, e.g. threshold-scanning techniques. Several experimental methods to measure the NLSP mass and lifetime are proposed and simulated in detail. We show that these methods can cover most of the lifetime range allowed by perturbativity requirements and suggested by cosmology in GMSB models. Also, they are relevant for any general low-energy SUSY breaking scenario. Values of c*tau_N1 as short as 10s of microns and as long as 10s of metres can be measured with errors at the level of 10% or better after one year of LC running with high luminosity. We discuss how to determine a narrow range (<~ 5%) for the fundamental SUSY breaking scale sqrt(F), based on the measured m_N1, c*tau_N1. Finally, we suggest how to optimise the LC detector performance for this purpose.
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