One-dimensional non-Hermitian quasicrystals with parity and time-reversal (${\cal PT}$) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and ${\cal PT}$-symmetry-breaking transition. This motivates us to investigate how the absence of ${\cal PT}$ symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve ${\cal PT}$ symmetry but, in most parameter regions, hosts a generic triple-transition scenario encompassing localization, topological, and degeneracy-breaking transitions. These transitions can be understood within the dual momentum-space representation featuring asymmetric couplings. Specifically, the localization boundary is obtained from the dual momentum-space representation by comparing the Lyapunov exponent, or inverse localization length, of the reciprocal Aubry-Andr'e model with the nonreciprocal hopping-asymmetry ratio of the dual lattice. The accompanying topological transition is characterized by a point-gap winding number defined with respect to a reference energy inside the complex spectral gap, whereas the degeneracy-breaking transition is understood from the loss of the reciprocal (n-n) spectral symmetry. The system also exhibits a localization-delocalization transition analogous to that in the Hermitian case, which occurs without accompanying topological or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in ${\cal PT}$-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.