Subj.
I cannot use singlelinecheck=off
because it aligns it to the left border of the page.
And why table is too narrow?
\documentclass[12pt,a4paper,oneside]{article}
\usepackage[left=35mm,right=15mm,top=2cm,bottom=2cm,bindingoffset=0cm]{geometry}
\usepackage[utf8x]{inputenc} % указать кодировку русского текста
\usepackage[russian]{babel} % указать, что язык текста - русский
\usepackage{cmap}
\usepackage{makeidx}
\makeindex
\usepackage{indentfirst}
\setlength{\parindent}{1.25cm}
\usepackage{caption}
\captionsetup[table]{labelsep=endash}
\usepackage{ltablex}
\usepackage{lipsum}
\begin{document}
\keepXColumns
\begin{tabularx}{\textwidth}{| l | X |}
\caption{A long description of this awesomest table ever. No, seriously.}\\
\hline
First & Second\\ \hline \hline
\endfirsthead
\captionsetup{singlelinecheck=off}
\caption{A short one}\\
\hline
First & Second\\ \hline
\endhead
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
Lols & \lipsum[150] \\ \hline
\end{tabularx}
\end{document}