it seems to mean something strong as disagreement, near to the italian "FARE SCHIFO" - about "how bad the station of shinjyuku HAS BEEN MESSED UP", "what HORRID CONDITIONS it is in"....hope this can help you,
Yes, futaro, it's "experience." Its original meaning is, as Flaminius has written, "spots （目）on a dice." When we say 今日ひどいめに会った, one doesn't associate め with eyes, but rather spots on a dice. Even if め in the sense of "spots on a dice" derives from "eyes", nobody think it that way when breaking down the said sentence.