Recent experiments revealed non-Fermi-liquid resistivity in the unconventional superconductor $Sr_2RuO_4$ when strain pushes one of the Fermi surfaces close to a van Hove singularity. The origin of this behavior and whether it can be understood from a picture of well-defined quasiparticles is unclear. We employ a Boltzmann transport analysis beyond the single relaxation-time approximation based on a single band which undergoes a Lifshitz transition where the Fermi surface crosses a van Hove singularity either due to uniaxial or epitaxial strain. First, analytically investigating impurity scattering, we clarify the role of the diverging density of states together with the locally flat band at the point of the Lifshitz transition. Additionally, including electron-electron scattering numerically, we find good qualitative agreement with resistivity measurements on uniaxially strained $Sr_2RuO_4$, including the temperature scaling and the temperature dependence of the resistivity peak. Our results imply that, even close to the Lifshitz transition, a description starting from well-defined quasiparticles holds. To test the validity of Boltzmann transport theory near a van Hove singularity, we provide further experimentally accessible parameters, such as thermal transport, the Seebeck coefficient, and Hall resistivity and compare different strain scenarios.