Did Lincoln say Prohibition will work great injury to the cause of temperance?

No.

See _They Never Said It: A Book of Fake Quotes, Misquotes, and Misleading Attributions_ by Paul Boller (Oxford University Press, 1989).

One way you can easily spot these sorts of fake quotes from prominent historical figures is to increase your level of general historical knowledge. That lets you realize quickly when something is unlikely given the context of the time of the quote.

In this case, Prohibition as a movement was very much in retreat and decline when Lincoln was President. It didn't become nationally significant again until decades after his death. (And Lincoln as President had certain issues that essentially dominated his agenda to the exclusion of all others, notably the small matter of the Civil War.)

Another example of a former President claimed to have said something that is incongruous in the context of the time is the fake quote of Jefferson warning that corrupt corporations and banks will use inflation to dispossess the common people of America, who will "wake up homeless on the continent their Fathers conquered."

While inflation is in truth a terrible danger to ordinary workers and retirees, and banks often do not have the best interests of those people at heart, consider these points:

[a] the technical term "inflation" was not in use yet in the literature of political economy in Jefferson's day. (Political economy was the term used by Jefferson and his contemporaries to describe much of what we would today call "economics", economics as a distinct, rigorous, systematized field not yet really having come into clearly defined existence back then.)

[b] The continent of North America had not yet been "conquered" by the United States at the time of Jefferson's death. Considerable amounts of territory that are now part of the contiguous United States still lay outside of American ownership and control during Jefferson's life, and it was by no means clear that those areas would ever be part of the USA.

Those clues should make it clear that something's not right with the attribution. And the Jefferson quote does in fact not appear anywhere in any of his archived writings or books. Someone, somewhere, simply made it up.

Here's one more quick example of how not to be hoodwinked:

Barack Obama's Vice President, Joe Biden, interviewed by CBS News during the 2008 campaign, said this:

"When the stock market crashed, Franklin Roosevelt got on the television and didn't just talk about the princes of greed. He said, 'look, here's what happened.'"

Again, if you have a solid fund of general historical knowledge, you can immediately identify that, while the Biden quote is quite real, Biden is proving himself to be out of his intellectual depth.

[a] The stock market crashed in 1929, under FDR's predecessor Herbert Hoover. Roosevelt didn't take office until 1932.

[b] Television existed in FDR's day, but its technical state was incredibly primitive, and deployment was incredibly limited. There were only a few hundred sets in the entire country, and Roosevelt never "got on the television" to address Americans -- what would have been the point? The ubiquitous mass media of FDR's day was radio, on which he made many regular addresses to the nation.