A good strategy in such problems is to massage the problem into recognizable limits. (EDIT: I like this approach mainly because it avoids Taylor expansion and l'Hôpital's rule. This is, however, not the simplest approach.)
We can "simplify" given function as follows: $ \begin{eqnarray*} \frac{\frac{1}{x} - \frac{1}{e^x - 1} - \frac{1}{2}}{x} &=& \frac{(2-x)(e^x - 1) - 2x}{2x^2 (e^x - 1)} \\ &=& \frac{\color{Blue}{(2-x)}(e^x - 1)- \color{Blue}{(2-x)} \frac{2x}{2-x}}{\color{Red}{2} \ \color{Magenta}{x^2} \color{Green}{(e^x - 1)}} \\ &=& \frac{\color{Blue}{2-x}}{\color{Red}{2}} \cdot \frac{\color{Magenta}{x}}{\color{Green}{e^x - 1}} \cdot \frac{e^x - 1 - \frac{2x}{2-x}}{\color{Magenta}{x^3}}. \tag{1} \end{eqnarray*} $ The first two factors both approach $1$ as $x \to 0$. Let us concentrate on the third factor. By Taylor expansion (or the formula for summing a geometric series), we have (for $|x| < 1$), $ \begin{eqnarray*} \frac{2x}{2-x} = \frac{x}{1 - x/2} &=& x + x \left(\frac{x}{2} \right) +x \left(\frac{x}{2} \right)^2 + x \left( \frac x 2 \right)^3 + \cdots \\ &=& \color{Red}{x + \frac{x^2}{2}} + \color{Blue}{x \left(\frac{x}{2} \right)^2 + x \left( \frac x 2 \right)^3 + \cdots} \\ &=& \color{Red}{x + \frac{x^2}{2}} + \color{Blue}{x \left(\frac{x}{2} \right)^2 \frac{1}{1 - \frac x 2}} \quad\quad \text{(summing the GP)} \\ &=& \color{Red}{x + \frac{x^2}{2}} \color{Blue}{+\frac{x^3}{2(2-x)}}. \tag{2} \end{eqnarray*} $ (Though I used infinite GPs to obtain the final expression, one could verify it directly as well. In particular, the two expressions are equal for all $x \neq 2$, not just $|x| < 1$.) Plugging $(2)$ in $(1)$, we have $ \begin{eqnarray*} \frac{e^x - 1 - \frac{2x}{2-x}}{x^3} &=& \frac{e^x - 1 \color{Red}{- x - \frac{x^2}{2}} \color{Blue}{-\frac{x^3}{2(2-x)}}}{x^3} \\ &=& \frac{e^x - 1 \color{Red}{- x - \frac{x^2}{2}}}{x^3} \color{Blue}{-\frac{1}{2(2-x)}} \end{eqnarray*} $ Once again, the second term has an easy limit of $\frac14$. The first term $ \frac{e^x - 1 - x - \frac{x^2}{2}}{x^3} $ is also a standard limit. This limit can be evaluated using, say, the l'Hôpital's rule or using the Taylor expansion of $e^x$; it's value is $\frac{1}{3!}$. Plugging in both these limits, we can get the final answer to be $ \frac{1}{3!} - \frac{1}{4} = -\frac{1}{12}. $
Bonus! If you wish to avoid Taylor expansion and l'Hôpital's rule even further, I will mention an "elementary" ways to evaluate limits such as $ \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} \quad\text{and}\quad \lim_{x \to 0} \frac{e^x - 1 - x - \frac{x^2}{2}}{x^3} $ assuming that these limits exists! (I stress that this is not a complete proof; yet I present it because I find the technique interesting.)
I will show the idea for the first limit, and leave the second one as an exercise. Suppose $A \stackrel{\text(def)}{=} \lim \limits_{x \to 0} \frac{e^x - 1 - x}{x^2}$ exists. Then $e^x = 1 + x + A x^2 + o(x^2)$. Therefore, by squaring: $e^{2x} = (1 + x + A x^2 + o(x^2))^2 = \color{Red}{1} + \color{Blue}{2x} + \color{DarkGreen}{x^2 (1+2A)} + o(x^2) .$ On the other hand, making the substitution $x \to 2x$ in the definition of the limit, we have $e^{2x} = \color{Red}{1} + \color{Blue}{2x} + \color{DarkGreen}{4A x^2} + o(x^2)$.
Equating the dominant terms in these two expressions, we must have $\color{DarkGreen}{4A} = \color{DarkGreen}{1 + 2A}$, which gives $A = \frac{1}{2}$.