A recently proposed zoned neutral atom processor achieves high fidelity by segregating the execution spaces of 1-qubit and 2-qubit gates and shows its potential as a candidate for high-accurate quantum systems. However, we observe that na"ively applying a program structure designed for existing architecture to the zoned neutral atom processor may result in devoting most of the execution time to the zone-to-zone travel of atoms. To reduce the movement between zones, the quantum program could be rewritten so there is less interleaving of 1-qubit and 2-qubit gates. In this context, we introduce \textit{Mantra} (Minimizing trAp movemeNts for aTom aRray Architectures), which exploits (i) fountain-shaped CZ (controlled Z) chain, (ii) 1-qubit gateless ZZ-rotation protocol, and (iii) preemptive alignment of identical zoned gates. Our evaluations show that \textit{Mantra} reduces inter-zone movements by 68%, decreases physical gates by 35%, and enhances the circuit fidelities by 17% over the standard program execution.