Let $x_n$, $x$ be nondecreasing cadlag functions on $[0,T]$. To get $x_n\to x$ in Skorokhods $M_1$-topology, we only have to prove convergence on a dense subset of $[0,T]$ including 0. (see Whitt: Stochastic Process Limits, Corollary 12.5.1).
Is there also a simplification (of a similar type) for convergence in the $J_1$-topology? I know that the same result is wrong for obvious reasons, but unfortunately I couldn't find any result with the "standard" $J_1$-Skorohod topology (at least in Whitts awesome book).
You can find some information on $J_1$-tightness of monotone cadlag processes in Silvestrov, "Limit Theorems for Randomly Stopped Stochastic Processes", Section 3.3.4.
There the author describes a criterion for $x \in D[0,T]$ non-decreasing which involves the sucessive moments of increments of size at least $\delta$ and for each $r \geq 1$ the minimal distance between the first $r$ sucessive moments. The tightness criterion is built upon this sequence of minimal distances (in addition to the typical boundedness property).
You can use this tightness criterion in order to strengthen the convergence from $M_1$ to $J_1$. In other words, first show that $x_n$ is converging pointwise on a dense subset including $0$ and $T$ (thus converging in $M_1$) and the verify the tightness property of Silvestrov (for a deterministic process).