This paper studies the problem of robust finite-time $H_\infty$ control for a class of uncertain discrete-time semi-Markov jump systems. The semi-Markov kernel approach is employed to model the semi-Markov jump process, in which the probability density function of sojourn time takes into account both the present and next system mode, and different types of sojourn time probability distribution can be considered in the proposed criteria. The method of estimating the maximum number of jumps is proposed for the finite-time analysis of the stochastic switching system. The upper and lower bound of sojourn time are considered simultaneously, which gives a novel concept of finite-time boundedness and guarantees the corresponding criteria based on the semi-Markov kernel to be numerically testable. On the basis of this concept, the $H_\infty$ performance is studied and the corresponding design method of the mode-dependent state-feedback control law is given to guarantee the corresponding closed-loop system to be finite-time bounded with given $H_\infty$ performance. Two numerical examples are delivered to illustrate the feasibility and effectiveness of the proposed theoretical method.