Full-potential all-electron density-functional calculations with mixed basis augmented plane wave plus local orbital--linearized-augmented plane wave have been carried out to investigate the electronic and geometric properties of bulk FCC Am and quantum size effects in the surface energies and the work functions of the (111) ultrathin films up to seven layers at six levels of approximations, namely, non-spin-polarized--no-spin-orbit coupling, non-spin-polarized--spin-orbit coupling, spin-polarized--no-spin-orbit coupling, spin-polarized--spin-orbit coupling, antiferromagnetic--no-spin-orbit coupling, and antiferromagnetic--spin-orbit (AFM-SO) coupling. The antiferromagnetic state including spin-orbit coupling is found to be the ground state of FCC Am with the $5f$ electrons primarily localized. Our calculations show that both spin-polarization and spin-orbit coupling effects play important roles in determining the equilibrium lattice constant, the bulk modulus, as well as the localized feature of $5f$ electrons for FCC Am. Our calculated equilibrium lattice constant of $9.21\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and a bulk modulus of $25.64\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ at the AFM-SO level are in good agreement with the experimental values of $9.26\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and $29.4\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, respectively. For the films, surface energy rapidly converges and the semi-infinite surface energy is predicted to be $0.81\phantom{\rule{0.3em}{0ex}}\mathrm{J}∕{\mathrm{m}}^{2}$ at the ground state. The work function of the FCC Am(111) surface at the AFM-SO ground state is predicted to be $3.06\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, and the work function shows some oscillations when the number of layers is less than five, while it becomes relatively stable when the number of layers is greater than five. This suggests that a five-layer film may be required for any future adsorption investigation that requires, for example, an accurate prediction of one-electron properties and adsorbate-induced work function shift.