We study the Landau-Ginzburg mirror of toric/non-toric blowups of (possibly non-Fano) toric surfaces arising from SYZ mirror symmetry. Through the framework of tropical geometry, we provide an effective method for identifying the precise locations of critical points of the superpotential, and further show their non-degeneracy for generic parameters. Moreover, we prove that the number of geometric critical points equals the rank of cohomology of the surface, which leads to its closed-string mirror symmetry due to Bayer's earlier result. Contents 1. Introduction 1 2. The SYZ mirror of a log Calabi-Yau surface 6 3. Critical points of Laurent Polynomials 16 4. Singularities of mirror Landau-Ginzburg potentials 20 5. Homological Mirror Symmetry 43 Appendix A. Estimates for the second-order expansion of W 50 References 52

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