How is the MS in finance program at University of Buffalo School of Management as per career opportunities in the USA as an Indian student?
Can I still get into it with 3-year bachelor's degree from India (BBA-69% from University of Pune)?

Ask the school. Many students end up working outside the US, I think: https://management.buffalo.edu/degree-programs/master-of-science-ms/finance/career-preparation/employment.html 
If you have a competitive GMAT or GRE, they may be flexible on the degree. 

US schools tend to be more forceful with their 4-year degree requirements, but yes, you should ask the school directly.

As far as career opportunities are concerned, you'd have a couple of issues counting against you. One, this is a finance degree, which may be a bit restricting in terms of the *kinds* of jobs you'd have a good chance at getting.

Secondly, compared to other, more well known schools, this one is certainly more regional. The job opportunities will be regional. And I'm not talking all of New York State, I'm talking jobs in upstate New York.

I'm not sure how many finance jobs are there, and how many would be interested in hiring an international employee with an uncertain future in the US (given the H1B issues), but probably not a lot.