Is there any other way to access the n-th element of a BlankSequence x__?

You can also use Slot (that is, #n & @ x):

Cases[{x__, 1} :> ( #4 & @ x)][
  {{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w,  1}, {a, b, c, 1}}]

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and Indexed (that is, Indexed[{x}, n]):

Cases[{x__, 1} :> Indexed[{x}, 4]][
  {{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w, 1}, {a, b, c, 1}}]

enter image description here

compare with Part:

Cases[{x__, 1} :> {x}[[4]]][
  {{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w, 1}, {a, b, c, 1}}]

enter image description here

You can use Repeated instead of BlankSequence to match the fourth element directly. Using the example from kglr:

Cases[{Repeated[_, {3}], x_, ___, 1} :> x][
  {{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w,  1}, {a, b, c, 1}}]

{d, w}

Are you looking for this sort of result?

{{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w, 1}, 
 {a, b, c, 1}} /. {x__, 1} :> Last[{x}]

(* Out: {{a, 2}, e, w, c} *)

Or:

{{a, 2}, {a, b, c, d, e, 1}, {x, y, z, w, 1},
  {a, b, c, 1}} /. {x__, 1} :> Take[{x}, {Length[{x} - 1]}]

(* Out: {{a, 2}, {e}, {w}, {c}} *)

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