Maulana Azad National Urdu University

Hi,
Yes, ofcourse you can go for BA in Arabic.

Yes, the medium of education is Urdu. For details visit: https://www.shiksha.com/b-tech/course/b-tech-in-computer-science-maulana-azad-national-urdu-university-hyderabad-269113

Yes, you are eligible for MA in islamic after completing your B.Sc.

For admission in MA in Arabic, you need to have 45% in graduation. The candidate must have studied Arabic as the main language in graduation or equivalent.
The candidates with advance/senior diploma in Arabic with above basic qualifications are also eligible. Those with Arabic as the main language will be preferred.

Admissions to BA history is based on merit basis you can apply online for visiting the college website.
For more detail information visit the below college link:
http://www.manuu.ac.in/Eng-Php/admission2016-17.php
Hope this helps you.

Eligibility required for admission:
1.Minimum 45% required in graduation.
2.Candidate must have studied Urdu as main language or optional subject in Degree or Equivalent examinations.
2.Candidates from affiliated subjects i.e. Arabic & Persian are also considered for admission.
*Admission is based on the marks obtained in Graduation.

Total 50 seats are for B.Ed in MANUU.

Even your graduation is other than Islamic History. Still, you are eligible to apply for MA in Islamic History course from MANUU as basic requirements for this course is 40% marks in any Bachelors degree or equivalent degree from recognized university.

Few prominent ones are:
1. Indian Rural Management Institute, Ahmedabad (IRMA)
2. National Institute of Rural Management (NIRM)
3. Tata Institute of Social Sciences (TISS Tuljapur campus)
4. All Agricultural Universities in India
5. National Council of Rural Institutes (NCRI)
6. National Institute of Rural Development (NIRD)
7. Xavier Institute of Social Sciences (XISS)