Here you go:
The missing bit was you needed zero-indexed columns as Cint instead of 1 and 2.
using JuMP
using Gurobi
column(model, x) = Cint(Gurobi.column(backend(model), index(x)) - 1)
model = direct_model(Gurobi.Optimizer())
@variable(model, x >= 0.001)
@variable(model, y <= 2)
@objective(model, Max, x)
# y = log(x)
GRBaddgenconstrLog(backend(model), "logCost", column(model, x), column(model, y), "")
optimize!(model)
value(x), value(y)