Developing quantum algorithms adaptive to specific constraints of near-term devices is an essential step towards practical quantum advantage. In a recent work [Phys. Rev. Lett. 131, 103601(2023)], we show cold atoms in an optical cavity can be built as a universal quantum optimizer with programmable all-to-all interactions, and the effective Hamiltonian for atoms directly encodes number partitioning problems (NPPs). Here, we numerically investigate the performance of quantum annealing (QA) and quantum approximate optimization algorithm (QAOA) to find the solution of NPP that is encoded in the ground state of atomic qubits. We find the success probability of the standard QA decays rapidly with the problem size. The optimized annealing path or inhomogeneous driving fields only lead to mild improvement on the success probability. Similarly, the standard QAOA always gets trapped in a false local minimum, and there is no significant performance improvement as we increase the depth of the quantum circuit. Inspired by the counterdiabatic driving, we propose an adaptive ansatz of QAOA which releases the parameter freedom of the NPP Hamiltonian to match higher-order counterdiabatic terms. Through numerical simulations, we find that our adaptive QAOA can achieve the optimal solution within very small circuit depth. It is thus worth paying the extra optimization cost of additional parameters for improving QAOA performance. Therefore, our adaptive QAOA provides a promising choice for programmable atom-cavity systems to demonstrate competitive computational power within its quantum coherence time.