M.A

Hey
You will have to go through the prospectus of the course provided by the college to know more about the admission procedure, fees, eligibility, scholarships etc.
https://www.shiksha.com/university/manuu-maulana-azad-national-urdu-university-hyderabad-25139

Dear Nadeem,
Yes definitely you can do MA in psychology.
Eligibility for enrolling in M.A. (Psychology) in most of the top universities is pass with 50% aggregate marks in Bachelor's Degree.
People from psychology and non-psychology both can become a career counselor.
There are lot of certification programs are available to become a career counselor.
You have 2 ways
1. Complete MA in psychology then go for career counsellor certification program
2. Any graduation and then go for career counsellor certification program
There is no eligibility for career counsellor certification program. Refer below link
https://certifiedcareeranalyst.com/

Different colleges have different fees. government colleges and private colleges have huge fees difference.
You will have to go through the prospectus of the course provided by the college to know more about the admission procedure, fees, eligibility etc.
https://www.shiksha.com/university/dr-c-v-raman-university-bilaspur-31270

Yes, you can definitely go for a psychology course or degree after completing your B.A. Although be sure about what you wish to do after your psychology Masters, in the sense of being a counselor, therapist and there are various other options in the field of psychology.