The victim, identified as Mosharraf Hossain, was admitted to Dhaka Medical College Hospital around 6:45pm Monday.
Our Narayanganj correspondent Tanveer Hossain confirmed that Mosharraf is known as a loyal supporter of the MP.
The office is on the second floor of 37/2, Box Culvert Road in Purana Paltan.
Paltan police Officer-in-Charge Rafiqul Islam said the shooting took place between 5:30pm and 6pm.
Four people had been taken to the police station for questioning about the incident, he said.
ASP Kamrujjaman Hira of DMP Paltan zone, who took the injured man to the hospital, said Mosharraf had claimed to be a Chhatra League activist.
The MP was not at the office during the shooting.