The Advanced Particle-astrophysics Telescope (APT) is a mission concept of a next-generation space-based gamma-ray and cosmic-ray observatory. We present the simulation of the instrument performance of the Antarctic Demonstrator for APT (ADAPT), a proposed long-duration balloon instrument based on a small portion of the full APT detector. We construct a semianalytical model of the MeV-GeV background for ADAPT based on observations from previous high-altitude balloon experiments and simulations of the upper atmosphere. We find that the ADAPT background is dominated by the gamma-ray albedo of the earth's atmosphere. In the presence of this background, we simulate a detector design based on a 45 cm $\times$ 45 cm detector composed of 8 thin layers of CsI:Na scintillators. We develop and optimize reconstruction algorithms for gamma-rays from a few hundreds of keV up to a few GeV energies. We present results of a complete off-line analysis to derive the best reconstruction methods. At photon energies from 30 MeV to a few GeV, ADAPT could provide degree-level to sub-degree-level observations of galactic and extragalactic gamma-rays with an effective area of above 0.05 $\mathrm{m}^2$. In the MeV regime, our simulation shows that ADAPT can achieve a degree-level localization accuracy for gamma-ray bursts down to about 1 MeV/$\mathrm{cm}^2$ in the presence of the gamma-ray and cosmic-ray background. ADAPT would be able to detect a few GRBs during the planned Antarctic balloon flight.