Could I compute predictive density using paths of forecasts (similar to monte carlo method).
I have 1200 paths for each horizont of forecast, could I compute the mean and variance from the 1200 paths, and then evaluate the normal density (with this moments) in data observed?
What you describe sounds sensible. But your description is not sufficient to judge the correctness. Where do the paths come from, i.e. what is the type of uncertainty that you are sampling from? Parameter uncertainty? Initial State uncertainty?