We present simulations of the instrument performance of the Advanced Particle-astrophysics Telescope (APT), a mission concept of a $\gamma$-ray and cosmic-ray observatory in a sun-Earth Lagrange orbit. The key components of the APT detector include a multiple-layer tracker composed of scintillating fibers and an imaging calorimeter composed of thin layers of CsI:Na scintillators. The design is aimed at maximizing effective area and field of view for $\gamma$-ray and cosmic-ray measurements, subject to constraints on instrument cost and total payload mass. We simulate a detector design based on $3$-meter scintillating fibers and develop reconstruction algorithms for $\gamma$-rays from a few hundreds of $\mathrm{keV}$ up to a few $\mathrm{TeV}$ energies. At the photon energy above $30~\mathrm{MeV}$, pair-production/shower reconstruction is applied; the results show that APT could provide an order of magnitude improvement in effective area and sensitivity for $\gamma$-ray detection compared with the Fermi Large Area Telescope (LAT). A multiple-Compton-scattering reconstruction at photon energies below $10~\mathrm{MeV}$ achieves sensitive detection of faint $\gamma$-ray bursts (GRBs) and other $\gamma$-ray transients down to $\sim0.01~\mathrm{MeV/cm}^2$ with degree-level to sub-degree-level localization accuracy. The Compton analysis also provides a measurement of polarization where the minimum detectable degree of polarization for $\sim1~\mathrm{MeV/cm}^2$ GRBs is below $20\%$. In addition to the APT simulations, we present the simulated performance of the Antarctic Demonstrator for APT, a 0.5m-square cross section balloon experiment that includes all of the key elements of the full APT detector.