Congruences arising from Ap\'ery-type series for zeta values

Recently, R. Tauraso established finite $p$-analogues of famous Ap\'ery series for $\zeta(2)$ and $\zeta(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Ap\'ery-type series for $\zeta(4)$ and $\zeta(5).$ We also prove a $p$-analogue of Zeilberger's series for $\zeta(2)$ confirming a conjecture of Z. W. Sun.