In this paper, we consider sequential representations of the recently introduced $r$-Jacobi-Stirling set partitions (denoted by $P(n,k)$) and study various statistics on these representations. We compute an explicit formula for the generating function which counts members of $P(n,k)$ where $k$ and $r$ are fixed according to these statistics in the case of levels, descents and ascents. In each case, we use a more-or-less uniform strategy which also yields the distribution of the statistic on those members of $P(n,k)$ ending in a certain letter. Finally, we give explicit formulas for the total number of levels, descents and ascents within all of the members of $P(n,k)$, providing both algebraic and combinatorial proofs.