Distance

Doing distance MBA program will be of no use,you will be wasting your time,money will be earned but the time will never come back.
Go for regular Program.goodluck

You can check the UGC website for their approval, if this institute is listed under UGC for offering courses under ODL mode, then you can apply. If it is not listed then check again on DEB sites and confirm. If they have not given permission to offer the courses, then don't join. Moreover, UGC has certain norms regarding offering of courses under distance mode, there is an issue for territorial jurisdiction also. So, check carefully before joining such institute.

Yes, you can pursue MBA through distance from any UGC recognised university.

Marketing is the most recommended and accepted.
https://www.shiksha.com/mba/colleges/distance-correspondence-mba-colleges-in-hyderabad