Answer
$1.00433$
Work Step by Step
We have 158 dark fringes shift, and we know that one fringe shift corresponds to a $\lambda$ path length change.
In our case, the length of the glass container does not change but the number of wavelengths does.
The length of the container is $L$, so the number of wavelengths when it is empty (vacuumed) is given by
$$N_{1}=\dfrac{L}{\lambda}$$
and the number of wavelengths when it is filled with gas is given by
$$N_{2}=\dfrac{L}{\lambda_{n_{gass}}}=\dfrac{Ln_{gas}}{\lambda}$$
We know that the light moves inside the container twice forth and back when reflected from the mirror; so the number of dark fringes shifts is given by
$$N=158=2\left[N_2-N_1\right]$$
Plugging from the previous two formulas;
$$158=2\left[\dfrac{Ln_{gas}}{\lambda}-\dfrac{L }{\lambda}\right]$$
Solving for $n_{gas}$;
$$\dfrac{158}{2}= \dfrac{Ln_{gas}}{\lambda}-\dfrac{L }{\lambda}=\dfrac{L(n_{gas}-1)}{\lambda} $$
Thus,
$$n_{gas}=\dfrac{158\lambda}{2L}+1 $$
Plugging the known;
$$n_{gas}=\dfrac{158\times 632.8\times 10^{-9}}{2\times 1.155\times 10^{-2}}+1 =\color{red}{\bf 1.00433} $$
