So there are a lot of problems with that quote, lets start from the beginning.

And Abraham Lincoln developed a phobia. He couldn’t beat Robert E. Lee.

By Grant's appointment to Chief of the Union Army in March 9th, 1964, Lee had already lost 2 major pitched battles. The first was Antietam on September 17, 1862 against George McClellan, the second was Gettysburg on July 2nd-July 4th 1863 against George Meade. The Army of the Potomac had beaten the Army of Northern Virginia both times when it tried to attack north in to Union territory, what it hasn't done is make a successful push towards Richmond.

And one day, it was looking really bad. And Lincoln just said, 'You,' hardly knew his name.

Obviously false, Lincoln knew Grant (certainly by name) long before March 1864. While Lincoln admitted to not knowing if he met grant in person, he certainly addressed Grant by name in a letter to grant at the beginning of the siege of Vicksburg ^1.

And they said, 'Don’t take him, he’s got a drinking problem.' And Lincoln said, 'I don’t care what problem he has. You guys aren’t winning.'

Grant's drinking problem was widely reported by the media at this time. However, the true extent of his drinking is far less serious. Grant simply did not like the reporters that followed his Army, and the reporters in turn exaggerated his drinking for the audience back home. Everyone "knew" Grant was a stumbling drunkard when he more of a very high functioning alcoholic. To quote biographer Geoffery Perret.

The entire staff, as well as most of Grant’s division and corps commanders, was well aware of his drinking problem. [Brig. Gen. John A.] McClernand tried to make capital out of it and one or two other officers expressed their disgust at Grant’s weakness, but to the rest, it did not matter. A few were alcoholics themselves, but the main reason it was tolerated was that when Grant got drunk, it was invariably during quiet periods. His drinking was not allowed to jeopardize operations. It was a release, but a controlled one, like the ignition of a gas flare above a high-pressure oil well ^2

Reports (true or not) of Grant being publicly drunk were much less common after his victory at Vicksburg.

And his name was Grant. General Grant. And he went in and he knocked the hell out of everyone.

The problem here is that Grant didn't "knock the hell out of everyone" after he was given command over the Army of the Potomac. The Overland Campaign that Grant directed was notoriously bloody, and the raw casualty ratios were worse for the Union than the 1862 Peninsula Campaign under McClellan. Grant "won" in the end by refusing to withdraw after a tactical defeat like his predecessors, and continued to maneuver his army and force Lee to stretch himself thinner and thinner.

Grant's major battlefield victories were in the west, before he was appointed Commanding General of the Union Army; Fort Donelson, Shiloh, Vicksburg, and Chattanooga.

And he’s finally being recognized as a great general.

Vicksburg had made Grant famous before being transferred to the Eastern Theater, and his reputation only rose with the Chattanooga campaign. Grant was wildly popular right as he assumed command of all Union forces, but the bloody battles of the Overland Campaign greatly hurt his popularity.

Until March 1864, when Lincoln appointed Grant to be General-in-Chief of the entire Union Army, Lee had indeed pretty much ran rampant over the Union Army of the Potomac. The North's had only two real victories of note against the South's Army of the Northern Virginia, one at Antietam Creek in Maryland, and the other at Gettysburg. Lincoln was particularly incensed after Gettysburg as Union General Meade had a golden opportunity to destroy the Confederate Army while it was fleeing South, but failed to do so. Historians have speculated that the hesitation on Meade's part was due to Lee's legendary reputation that was well earned through the first 2 1/2 years of the war.

Grant, who had left the military before the Civil War, had risen quickly through the ranks in the Western theater of the war and had eventually taken command of the Union Army of Tennessee. He understood the advantages in manpower and resources the North had over the South and, more importantly, possessed the ability to exploit those advantages. His victory at Vicksburg coming at the same time as Lee's escape from certain destruction after Gettysburg, made Grant a logical choice for Lincoln to appoint General-in-Chief of all Union forces.

As far as Grant's reputation as a drunkard, it stems mainly from political infighting inherent of any military, and became more permanent by opponents of his after the war and his entrance into politics. The rumors began after the Battle of Shiloh, during which the combined 23,000+ casualties dwarfed the total number of casualties suffered by the United States in all other total wars at the time. Lincoln, impressed with Grant's resolve, refused any attempt to dismiss him, making the now famous comment, "I can't spare this man; he fights." There is no evidence that Grant was ever inebriated before or during any battle.

Sources: Sears, Stephen W. Gettysburg

Longacre, Edward G. General Ulysses S. Grant: The Soldier and the Man