Manipal

Manipal Institute of Technology under Manipal University offers M.Tech in Environmental Engineering and Construction Engineering & Management for Civil engineering graduates. You may refer to following link for further details: http://manipal.edu/mit/programs/program-list/mtech.html

You may do Ph.D from IGNOU or AIMA-AMU by distance learning program.

Dear Soumyajit,
Yes, you can pursue it. Sikkim Manipal University has got good reputation. All the best.

Somesh, admission to B.Tech lateral entry in Manipal is based on their entrance test which was on 2nd June and 7th June. Check the link below
http://manipal.edu/mit/programs/program-list/btech/btech-lateral-entry.html
There are many other good universities where admissions are still going on. Their admission criteria is LEET score or their own test or your marks in previous qualifying exams. Most of them offer good placements. Some even offer scholarships.
Check the websites of some good universities for more details.
Good luck.

Among your options, VIT is the best option because of its ranking, facilities, faculty and placement records.

If you're an NRI then you may not write the exam and simply apply through NRI applications but if you're not, you'll have to give the exam.