Ryan Elliot Giroux, 41, was being questioned Wednesday evening, Detective Esteban Flores of the Mesa Police Department said.
He has an extensive criminal background, Flores said. Court records show Giroux is on probation.
A spokesman for Banner Medical Center told CNN that Giroux was brought in by police, treated and released at 2:30 p.m. local time.
One of the victims who was wounded was in critical condition, the detective said.
The series of shootings took place Wednesday morning in several locations, Flores said.
The shootings began at a motel where the suspect got into an argument in a room then shot three people whom he likely knew outside, Flores said.
One man who was shot at the Tri-City Inn, across the street from the East Valley Institute of Technology
, died, Flores said.
Two women were wounded as was an EVIT student who was at a restaurant run by the school on the other side of the street, Flores said. The wounded man crossed the road to be treated when paramedics arrived.
The suspect carjacked a Honda and later shot a man after breaking into his apartment and another man in the apartment complex next door. Both men were expected to live. Flores said the suspect likely was trying to rob his victims as he tried to get away from authorities.
Flores said the suspect was found by Mesa's SWAT team in a second-floor condo that was vacant. Police moved into the unit and used a Taser on the suspect.
"Obviously he has tattoos on his neck and his face, his forehead. Those are typical tattoos that you do see sometimes on white supremacists, but we don't know if that has anything to do with what happened today," Flores said.
According to online inmate records, Giroux has been in prison twice for felonies that included an attempt to commit aggravated assault. His longest and most recent sentence was in July 2007 for seven years and six months, but he was released in October 2013.
He also was sentenced to two months in jail in 2014 for a probation violation of the 2007 case. His probation extends through October 2016.
The FBI and the Bureau of Alcohol, Tobacco, Firearms and Explosives assisted local authorities in the manhunt.
News of the shootings caught the attention of the state's senior U.S. senator.
"My thoughts and prayers are with the people of #Mesa right now #Arizona," U.S. Sen. John McCain said on Twitter Wednesday.
Mesa is about 20 miles east of Phoenix and is Arizona's third-largest city.