Born on a farm near Midway, Woodford County, Kentucky, he and his siblings inherited the property on his father's death.
Alexander was sent to study in England, where he earned a degree at Trinity College, Cambridge. While there Robert Alexander became the beneficiary of an extremely wealthy uncle's Scottish estate, an inheritance that made him one of the richest young men in the world. Accordingly, he lived at the inherited estate in Scotland but after nine years there, in 1849 he decided to return to Kentucky. Back home he set about establishing a stud farm and in the early part of the 1850s returned to Europe to spend two years studying the techniques of breeding farms in Germany, France, and England. Starting with 921 acres purchased from his family, Robert Alexander built his Woodburn Stud at Spring Station, Kentucky into the leading horse breeding operation in the United States.
Alexander purchased African-American slaves Ansel Williamson and Edward D. Brown, who were taught the business of breeding and training horses. Both became horse racing trainers and would have outstanding careers. After being given their freedom, they both remained as employees of Alexander until his death. They went on to train Kentucky Derby winners and are both inductees of the National Museum of Racing and Hall of Fame.
In February 1865, soldiers of the Confederate Army attacked the village of Midway. They burned down the railroad station, robbed its residents, and stole fifteen of Alexander's prized thoroughbred horses.
Alexander died on December 1, 1867 and was buried in Frankfort Cemetery in Frankfort, Franklin County. His brother, Alexander John Alexander (known as "A.J."), took over the management of Woodburn Stud and further enhanced its reputation.