Yes, I discuss this concept in several places--but I guess it didn't make it into the index. I discuss cofinality in connection with the observations that aleph_omega being singular, whereas omega is regular, and we discuss that beth_omega is a singular strong limit cardinal, as a way of introducing the inaccessible cardinals, which would be an uncountable regular strong limit. The concept also comes up in the discussion of the orders of infinity, pointing out that no countable sequence of functions is cofinal in the order of eventual domination. Your observation about monotone maps from the ordinals is related to a similar observation about continuous functions from the long line to the reals: they must be eventually constant! Can you see the proof?