Set $y$ is given such that,
$$y= (y_1,y_2,...,y_n)$$ And $$y_1\leq y_2\leq...\leq y_n$$
Now take $y'$ such that $$y'= (y_1,\ y_2,...,\ y_j+c,\ y_{j +1},....,\ y_k-c,\ y_{k +1}...,y_n)$$
Essentially subtracted a constant from some $y_j$ term and added the same constant constant to larger $y_k$ term. And it is given that even after doing this, $$y_j+c< y_{j+1} <...< y_{k-1}< y_k-c$$
What will be the relation between the variance of the set $y$ and $y'$? Answer is that the variance will reduce after the operation.
I have tried doing the problem by comparing the terms that change in the variance. This is because the mean remains same for both the set. But all efforts in vain since I am not getting anywhere. Please help.
Your approach to compare the terms that change in the variance is exactly right.
Subtracting the old terms from the new terms yields (a multiple of)
$$ (y_j+c)^2+(y_k-c)^2-y_j^2-y_k^2=2c(y_j-y_k+c)\;. $$
It is given that $y_j+c\lt y_k-c$ and thus $y_j-y_k+c\lt -c$. You didn't state that $c$ is positive, but we need to assume that it is to conclude that the difference is negative and thus the variance is reduced. (The active form is “decreases”, not “reduces”.)