General solution for using diffrent heads and levels:
customMap[fn_, list_, heads_List, level_] :=
Replace[list, Alternatives @@ (Pattern[x, Blank[#]] & /@ heads) -> fn[x], level]
customMap[(# + 3) &, {a, 3 b, {c, d}, e}, {List}, {1}]
(*Out: {a, 3 b, {3 + c, 3 + d}, e} *)
Solution 1
Using customMap:
ListMap[fn_, list_, level_] := customMap[fn, list, {List}, level]
ListMap[(# + 3) &, {a, 3 b, {c, d}, e}, {1}]
(*Out: {a, 3 b, {3 + c, 3 + d}, e} *)
Solution 2
Using Replace:
ListMap[fn_, list_, level_] := Replace[list, x_List -> fn[x], level]
ListMap[(# + 3) &, {a, 3 b, {c, d}, e}, {1}]
(*Out: {a, 3 b, {3 + c, 3 + d}, e} *)