The priest, Youssef Adel, was driving to a church when gunmen in another car sped past him and opened fire in the central district of Karradah, two police officials said. The officials spoke on condition of anonymity because they weren't authorized to release the information.
Adel's religious affiliation was not immediately available. Iraq is a predominantly Islamic country but has a number of religious minorities, including Catholics and Orthodox Christians, who have faced frequent attacks.
The minibus was bombed shortly before 8 a.m. as people were going to work on busy Palestine Street in eastern Baghdad, a police officer said, declining to be identified for the same reason.
The four passengers killed and 15 wounded were primarily workers and vendors from the Sadr City district who were on their way to commercial areas elsewhere in the capital, the officer said.
The attacks underscored the dangers that continue to face Iraqis in Baghdad and elsewhere as attacks persist despite a sharp decline in violence over recent months amid a U.S.-Iraqi security crackdown.
The Iraqi government, meanwhile, eased security measures in two Baghdad neighborhoods that are strongholds of Muqtada al-Sadr's Mahdi Army militia - Sadr City and Shula - amid complaints of food shortages nearly a week after the radical Shiite cleric issued a cease-fire order.
Trucks carrying maintenance teams, food, oil products and ambulances are now allowed to get into the areas, according to a statement issued by Brig. Gen. Qassim al-Moussawi, Baghdad's chief Iraqi military spokesman.
The areas - including Sadr City, home to some 2.5 million Shiites and the militia's largest base - have suffered as a vehicle ban remained effect despite the lifting of a curfew elsewhere in the capital earlier this week.
While al-Sadr's order put an end to large-scale fighting that broke out over a government crackdown in the southern city of Basra, clashes have continued between his fighters and Iraqi security forces.
Sporadic gunfire could still be heard in Basra, although it was relatively calm as aid workers delivered humanitarian assistance to the beleaguered residents.
Associated Press Television News footage showed neighbors in the Hayaniyah area examining the rubble of a house they said was destroyed in an air strike Friday evening in the militia stronghold.
Police said five people were killed in the strike, acknowledging they included an unspecified number of militants who had fired a mortar at Iraqi security forces.
British military spokesman Maj. Tom Holloway said an attack helicopter hit a position from which militants were firing at Iraqi forces in Hayaniyah, but he gave no information about casualties and did not specify whether the aircraft was U.S. or British.
A series of air strikes have struck suspected militant positions since the fighting erupted on March 25, drawing American and British forces into the battle and casting further doubt on the ability of Iraqi forces to take over their own security.
The White House has conditioned further U.S. troop withdrawals on the readiness of the Iraqi military and police.
Prime Minister Nouri al-Maliki had suggested he planned similar crackdowns in Sadr City and Shula earlier this week, but instead he ordered a nationwide freeze on Iraqi raids against Shiite militants on Friday.
The reversal came after al-Sadr hinted at retaliation if Iraqi security forces continue to arrest his followers.
In Amman, Jordan, David Shearer, the U.N. humanitarian coordinator for Iraq, said an estimated 700 people were killed and more than 1,500 were wounded in fighting that erupted after the Basra crackdown. He said the figures could rise "as facts and the numbers become more clear."
U.S. and Iraqi authorities had insisted the Basra operation was not aimed at al-Sadr's powerful political movement but instead at ridding the streets of criminals and gunmen who had effectively ruled the city since 2005.
But al-Sadr's supporters believed the crackdown was aimed at weakening their movement before provincial elections this fall. Al-Sadr expects to score major electoral gains against Shiite parties that work with the Americans.