Factual error: When Monk visits the prison for the first time, he is searched, passed, and has a mini-panic attack because people have touched him. Sharona steps forward to comfort him, and she puts her arms around him. This is a very serious breach of procedure and it is not possible for the guards to miss it, because Sharona has not been searched. It's a common technique for trying to pass contraband called 'a touch through' - a person with an item to be smuggled into the prison (drugs, usually), passes them on to a person who has been searched and is therefore 'clean'. Monk would be taken back to the guard's station to be searched again. He would not be allowed leniency because the guards happen to know or to like him - visitor ingress and egress is always done under CCTV and the guard would know he was being watched - quite possibly by someone who doesn't know who Monk and Sharona are.
Plot hole: Death row prisoners cannot donate organs. No doctor participates in an execution and after the lethal injection the medical examiner must wait at least fifteen minutes to test for cardiac activity before pronouncing the prisoner dead. This is enough time for hypoxia to render the heart, lungs and kidneys useless for transplant. Apart from this, organs intended for transplant must be removed within minutes of the death of the donor. Prisoners are not executed in operating theatres.
Factual error: There are a number of unrealistic situations portrayed inside the prison: 1) An inmate scheduled to be executed in less than an hour would not be left so unattended as to require the trustee to yell for help when he went into distress, or to allow the inmate and trustee to exchange objects. 2) A chronic rule-breaker like Spyder, who had just "put a guy's head through a wall," would not have gotten just one day in solitary confinement with no other loss of privileges. 3) The neo-Nazis would not be allowed to wear swastika lapel pins. 4) Death row inmates are not served their last meal by unescorted fellow prisoners.
Continuity mistake: A fight breaks out in the prison laundry, and Monk crawls into a giant dryer to hide, pulling the door mostly shut after him. But in the very next shot of the fight raging on, the dryer door is visible in the background - wide open. Next shot after that, it's shut again, and Monk sneaks it open to peek out. (00:42:00)
Continuity mistake: The large blue Nazi/Swastika tattoo on the left side of the head Nazi's neck changes from being neat and symmetrical to uneven during the course of the episode. This is most noticeable when comparing the exercise-yard "you stole the watch" scene to the black-and-white "solving the case" flashback scene.
Continuity mistake: When Monk and Sharona talk to Sylvia in the prison library the second time, the books on her desk change in various shots. At first, you see 3 piles of books, all with blue books on top. In the next shot the blue book closest to her is gone. In the next shot, she's straightening a red book next to her, but it goes back to being crooked.