It is possible to characterize the $k$-th eigenvalue as the following min-max problem. 

$$\lambda_k = \inf_{V \subset C^2(M), \, dim(V) = k} \max_{f \in V \backslash \{0\}} \frac{\int_M |\nabla f|^2 \,dVol} {\int_M f^2\, dVol}$$

Intuitively, $V$ is the subspace spanned by the first $k$-eigenfunctions and the maximization picks out the largest value of the Raleigh quotient. 

To prove the inequalities in your question, fix a substance $V$ and bound the numerator and denominator in terms of $a$ and $b$. The power of $n$ appears because this is how the volume form scales under rescaling and the extra power of $2$ comes from the scaling of the gradient term. 

To obtain the first inequality, we consider the space $V$ spanned by the first $k$ eigenfunctions with respect to $g$. Then we can bound the Raleigh quotient with respect to $\tilde g$ and then pass to the infimum. For the second inequality, we instead use the space spanned by the first $k$ eigenfunctions of $\tilde g$. 

Also, there is no need to consider a Riemannian cover here so I’m not sure what role that plays.