2 of 5 Added a missing word and made several minor improvements.

I didn't become familiar with the term mensch until the 1970s, when I moved from Texas to the east coast (Maryland) for college. At the time I assumed that it was simply a regional term. However, the frequency of "a mensch" in Google Book search results suggests that the term's popularity in published writings has grown substantially since the 1950s. Here is the Ngram chart for "a mensch" (red line) and for "a real mensch" (blue line) for the period 1900–2005:

The Ngram results suggest that few occurrences of "a mensch" appeared in print prior to 1960—and indeed Google Books search results find no instances of "a mensch" as a Yiddish expression in an English-language text until the two that appear in the 1950s. The first occurrence is from Ignaz Maybaum, The Jewish Mission (1951):

Geiger's humanist imperative "Sei ein Mensch" (be truly human) meant what Mensch meant in the German language of the eleventh century, and what has been kept alive until to-day in the Yiddish word "Mensch". But the nineteenth-century German language derived the word Mensch from the enlightenment age which saw man only as homo sapiens and not as the child of God. To be a Mensch meant to Geiger to be merciful. Geiger's Jewish humanism is expressed in Micah's words: "It hath been told thee, O man, what is good, And what the Lord doth require of thee: Only to do justly, and to love mercy, and to walk humbly with thy God" (VI, 8).

The second instance is from Philip Roth, "Epstein," in Goodbye Columbus: And Five Short Stories (1959):

Epstein opened his mouth. His tongue hung over his teeth like a dead snake.

"Don't you talk," his wife said. "Don't you worry about anything. Not even the business. That';; work out. Our Sheila will marry Marvin and that'll be that. You won't have to sell, Lou, it'll be in the family. You can retire, rest, and Marvin can take over. He's a smart boy, Marvin, a mensch."

Lou rolled his eyes in his head.

More than a dozen Google Books matches for mensch appear in English-language sources during the 1960s, and the numbers have continued to increase sharply in the decades since then.

Another measure of the increasing awareness (in U.S. English) of mensch is its inclusion, since the Ninth Collegiate (1983) in the Merriam-Webster's Collegiate Dictionary series. This is significant because these dictionaries are by no means exhaustive in their inclusion either of new terms or of terms that are infrequently used. The Ninth Collegiate gives a surprisingly late first-occurrence date for the word in English:

mensch n {Yiddish, fr. G, man, human being, fr OHG mennisco; akin to ON mennska humanity} (1953) : a person of integrity and honor

But just as unexpectedly, the Eleventh Collegiate (2003), while retaining the Ninth Collegiate's etymology and definition, bumps back the first occurrence date almost a hundred years, to 1856.

In my view, mensch is rapidly becoming Americanized, which means that the advice to "act like a mensch" is more and more likely to be understood in parts of the United Sates outside its traditional strongholds of the Northeast and big cities. Hence we get un-self-conscious article ledes like this one from David Greenberg, "It's a Myth That Nixon Acquiesced in 1960" in the Los Angeles Times (November 10, 2000):

Despite thousands of contested ballots in Florida's Palm Beach County, a lot of people are calling on Al Gore to act like a mensch and concede the election to George W. Bush—as they contend Richard M. Nixon did in 1960 when he lost to John F. Kennedy amid rumors of fraud.

Many newspapers far from Los Angeles are signed up to share stories from the L.A. Times, so a story like this one is apt to be read by people in many parts of the country, exposing them to the phrase "act like a mensch."

Nevertheless, on the strength of other responses to this question, I would caution you not to expect mensch to travel well outside the United States into other parts of the English-speaking world.