A first-principles formalism is employed to investigate the effects of size and structure on the electronic and electrochemical properties of Au nanoparticles with diameters between 0.8 and 2.0 nm. We find that the behavior of the ionization potentials (IPs) and the electron affinities (EAs) as a function of cluster size can be separated into many-body and single-electron contributions. The many-body part is only (and continuously) dependent on particle size, and can be very well described in terms of the capacitance of classical spherical conductors for clusters with more the 55 atoms. For smaller clusters, molecule-like features lead the capacitance and fundamental gap to differ systematically from those of a classical conductor with decreasing size. The single-electron part fluctuates with particle structure. Upon calculating the neutral chemical potential mu(0) = (IP + EA)/2, the many-body contributions cancel out, resulting in fluctuations of mu(0) around the bulk Au work function, consistent with experimental results. The values of IP and EA changes upon functionalization with thiolated molecules, and the magnitude of the observed changes does not depend on the length of the alkane chain. The functionalization can also lead to a transition from metallic to non-metallic behavior in small nanoparticles, which is consistent with experimental observations.