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I want to sort (or sortby) the list. List contains uninitialized variable m and I can assume that m is a Natural number that is much larger then any constant.
Example input:
a = {-116*m, 0, 3 - 11*m, 1 - m, -20*m - 7, -m}
Example output:
{-116*m, -20*m - 7, 3 - 11*m, -m, 1 - m, 0}
My effort:
MAGICNUMBER = 1000000;
Sort[a, (#1 /. m -> MAGICNUMBER) < (#2 /. m -> MAGICNUMBER) &]
Your own method seems fairly effective but it can be simplified:
a = {-116*m, 0, 3 - 11*m, 1 - m, -20*m - 7, -m};
SortBy[a, # /. m -> 1`*^12 &]
{-116 m, -7 - 20 m, 3 - 11 m, -m, 1 - m, 0}
a[[ Ordering[a /. m -> 1`*^12] ]]
{-116 m, -7 - 20 m, 3 - 11 m, -m, 1 - m, 0}
These methods will also perform much better than your use of Sort because the default sort algorithm is used rather than custom pairwise ordering.
For your example input, which happens to be a list with elements in which m appears up to linearly, you can simply do this:
lst = {-116*m, 0, 3 - 11*m, 1 - m, -20*m - 7, -m};
lst[[Ordering[D[lst, m]]]]
(*{-116 m, -7 - 20 m, 3 - 11 m, 1 - m, -m, 0}*)
which basically gets rid of each term that does not contain m, then orders the rest according to the coefficient of m.
If you have things like m^2 though this needs a bit more work.
Sort[lst][[Ordering[D[Sort[lst], m]]]] would work.
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