The Rule versus RuleDelayed issue probably warrants closing this question. That is a basic distinction easily discoverable in the documentation and books and examples etc. However, there are nuances here that it might be worth discussing.
If you do this
g[x3, x2, x1] /. {g[arg___] :> ReverseApplied[g][arg]}
you need to be aware that the g expressions will be evaluated. To demonstrate, let's give g a definition:
g[a_, b_, c_] := h[a, b]
Now evaluate
g[x3, x2, x1] /. {g[arg___] :> ReverseApplied[g][arg]}
(* h[x3, x2] *)
Presumably, what we wanted to get was h[x1, x2]. So, to answer your question of "what is the correct way of doing this?", you should do what you already knew how to do: use ReverseApplied, but do it directly.
ReverseApplied[g][x3, x2, x1]
(* h[x1, x2] *)
There is no need to go through the whole replacement exercise, just apply ReverseApplied directly.
Now, if the expression g[x3, x2, x1] does not evaluate to something other than itself, then the suggested replacements based on RuleDelayed will work, but the simpler way to do it would be to simply apply Reverse:
(* I'll use gg to distinguish from the previous case *)
Reverse@gg[x3, x2, x1]
(* gg[x1, x2, x3] *)