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Due to the limited decoherence time of qubits in the Noisy Intermediate-Scale Quantum (NISQ) era, optimizing the size and depth of logical quantum circuits for efficient implementation on sparsely connected physical architectures is crucial. In this work, we extend the Approximate Token Swapping (ATS) algorithm by incorporating qubit priority and the size of permutation cycles to be routed. We refer to this algorithm as Priority-ATS (PATS), which also considers CNOT gate errors while constructing the routing schedule for the qubits. We provide theoretical justification for the superior effectiveness of PATS over ATS and experimentally demonstrate its ability to improve the fidelity of the output state. Furthermore, we use depolarizing error channels to model the noisy CNOT gates, where each adjacent qubit pair has a distinct error rate. By employing realistic error rates, we showcase the robust improvement in output fidelity when using PATS as opposed to a noise-oblivious ATS routing scheme.