Before Capone was sent to Alcatraz, he arrived to the Atlanta U.S. Penitentiary in May 1933. Many people believed that Capone charmed his way into special privileges. This video claims that he did. Whether he had special privileges or not, Capone was not doing very well. On arrival, Capone was diagnosed with Syphilis and Gonorrhea and was also dealing with a cocaine withdrawal. There's even some evidence that say he was slipping into insanity based on many of his incoherent letters. Because of his mental and physical illnesses, he became prone to many prison attacks. Fortunately, Capone was cellmates with "Red" Rudensky, who was actually a small-time criminal apart of Capone's mob. Red, and a few other unknown prisoners, protected Capone. This is where many others felt that Capone was receiving special treatment. There aren't many online sources that talk about this, but this book will go into great detail.
To answer your question, Capone did pretty well at the Atlanta Prison, aside from his mental/physical health. Many believe that it's because of his special treatment that his transfer was admitted.
Attorney General Homer Cummings along with Sanford Bates, the head of the Federal Prisons, made arrangements to send Capone to a facility where he would be unable to leverage the system.
Capone stayed at the Atlanta U.S. Penitentiary until being transferred to Alcatraz on August 22nd, 1934. Once he arrived to Alcatraz, things changed. He no longer had contact with the outside world, except for newspapers and letters, but even those were rare. He no longer had power. He didn't "run" things. Capone began to suffer from neurosyphilis which affected his mental health even more, and you can go into more detail by reading this book. Capone carried out his sentence until January 6th, 1939 and was transferred to the Federal Correctional Institution at Terminal Island in California. He received parole in November 16th, 1939.
Another source:
Before Capone was at Atlanta U.S. Penitentiary, he spent several months in Eastern State Penitentiary (August 1929-March 1930). The prison is now open for tours, and they have a reproduction of what his cell may have looked like. It seems quite nice in comparison to the standard impression you get from the place.
This being said, whether he actually got special privileges or just used his wealth to acquire the nicest things that were available to all of the prisoners is unknown. Of course the Warden claimed that he received no special treatment, but there's very little credible information either way. Apparently, record-keeping at ESP was so bad that we don't even know which cell he was actually in.