Impulse Response for an Anti Causal Linear Shift Invariant System
You've already found the solutions of the characteristic equation as $\lambda_1=\frac12$ and $\lambda_2=-\frac14$, so you know that the solution must have the form
$$y[n]=c_1\left(\frac12\right)^n+c_2\left(-\frac14\right)^n\tag{1}$$
For the anti-causal solution we know that $y[n]=0$ for  $n&gt;0$. The constants $c_1$ and $c_2$ are simply determined by making sure that the difference equation is satisfied for $x[n]=\delta[n]$, given that we require $y[n]=0$ for $n&gt;0$:
$$\begin{align}n=2:\quad&amp;&amp;&amp;-\frac18y[0]&amp;=0\\n=1:\quad&amp;&amp;-\frac14y[0]&amp;-\frac18y[-1]&amp;=0\\n=0:\quad&amp;y[0]&amp;-\frac14 y[-1]&amp;-\frac18y[-2]&amp;=3\end{align}\tag{2}$$
From the first equation in $(2)$ we see that the impulse response actually starts at $n=-1$, not at $n=0$. The constants $c_1$ and $c_2$ can be determined from the second and third equation, which simplify to
$$y[-1]=0\quad\textrm{and}\quad -\frac18 y[-2]=3\tag{3}$$
Plugging $(3)$ into $(1)$ finally gives the actual values of $c_1$ and $c_2$, which I'm sure you can figure out yourself. The final anti-causal solution has the form
$$y[n]=\left[c_1\left(\frac12\right)^n+c_2\left(-\frac14\right)^n\right]u[-n-1]\tag{4}$$