Yes, we can calculate the properties of gold, both atomic and bulk. In practice, you start with atomic gold. You have to include relativistic effects, so you'll likely use some implementation of the Dirac-Fock equations as your starting point. To include electron correlation, you'll probably use a multiconfigurational model. You could also use perturbation theory for that, but either way, it's a time-consuming calculation for high Z atoms.
Regardless, this has all been done years ago, and from those precise calculations, people have built approximations for the core electrons of the heavier elements known as pseudopotentials. These pseudopotentials are basically smoothed out mean fields of inner core electrons. This is because calculating the precise electron densities from a plane wave basis (the most convenient basis to use in solid state calculations with periodic boundary conditions) requires a very fine integration mesh, as the multielectron wavefunction is highly oscillatory near the nucleus. As it turns out, the smoothed out fields don't tend to make too much of a difference when calculating out valence electron properties such as band structure, color, tensile strength, phonon modes, etc. (in short, most of the properties you'd be interested in).
With a decent core pseudopotential in hand, all you have to worry about is the valence electrons and setting up the proper boundary conditions. DFT programs are good enough at this point to have implemented variable boundary conditions to optimize crystal structure fairly readily (although the calculations get a lot quicker if you know the symmetry of the crystal structure). After that, the valence electronic structure is a straightforward DFT calculation, and from there, you can calculate a whole host of other properties (photoresponse functions, phonon spectrum, electron-phonon coupling, etc).
Each one of these steps has undergone decades of research, development, and implementation, so it's pretty routine to calculate something like a band structure for gold. In terms of having to do a calculation on 1000 or 10 000 atoms, nanoparticles are in fact one area which is quite challenging for atomistic quantum simulations. Supercomputers can do it, but there's still a lot of research going on in this area. There's a lot of interest in nanoplasmonics and nanophotonics that is driving this work right now.