The MOI website has posted their Ramadan working hours on their website:

"The Ministry of Interior announced the official duty time throughout the Holy month of Ramadan as listed below.

The General Directorate of Borders Passport and Expatriates Affairs, Department of Traffic and Department of Nationality and Travel Documents will be working from 8.00 am to 1.00 pm in the morning and from 8.00 pm to 12 am in the evening.

The headquarter of the Department of Criminal Evidence and Information (CEID) will be working from 8.00 am to 1.00 pm while the fingerprint branches located at headquarter of the department and at Al Duhail district will be working from 8.00 am to 1.00 pm in the morning and from 1.00 pm to 5 pm in the evening.

The fingerprinting branch located at each of Musaimeer and Emigration building will be working in two shifts, one in the morning from 8 am to 1 pm and another one in the evening, starting from 8 pm to 12 am.

The exit permit section will be working 24 hours a day"

Source: http://www.moi.gov.qa/site/english/news/2009/08/23/20872.html