As I understand it, the main difference between Metropolis and Metropolis-Hastings is the the former requires a symmetric proposal
distribution and the latter does not. The acceptance criterion
used by M-H 'corrects' for the 'bias' inherent in using an asymmetric distribution for moves to the next proposal.
In particular, I doubt that exponential moves you propose would work well even for M-H. You need to be able to move relatively freely
among possible proposals, Even if the exponential is offset so
that movement in both directions is possible, I think the M-H
acceptance criterion would essentially have to reject for a very
high proportion of jumps resulting from the long exponential tail.
Addendum (7/19): For archival purposes, I should note that
the main applications of both Metropolis and M-H methods is to
simulate from multivariate distributions, where choices of methods are often quite limited. Based on what I saw
in the reference you provide, this is a drill problem to sample
from a univariate distribution. There are much simpler acceptance-rejection methods for univariate cases that produce iid sequences (for example, see my answer to question 1358708 on this site).
Also, of course, sampling from the distribution of your reference
is most efficiently done by a direct method that does not use
acceptance-rejection at all. (I am not at all objecting to this
kind of drill exercise as a way towards understanding Metropolis and M-H;
I have assigned similar problems myself. But no one should mistake
Metropolis or M-H as an optimal sampling method in the current context.)