In this article, the generalized Apostol type-Sheffer sequences are introduced and their properties including the quasi-monomiality, determinant form and series and conjugate representations are derived via Riordan array techniques. The generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi-Sheffer sequences are considered as their special cases. Certain examples are framed in terms of the generalized Apostol Bernoulli-associated Laguerre sequences, generalized Apostol-Euler-Hermite sequences and generalized Apostol-Genocchi-Legendre sequences to give the applications of main results. The numerical results to calculate the zeros and approximate solutions of these sequences are given and their graphical representations are shown