There are sports and inter-college tournaments facilities from Mysore Medical College. Gym can be in the campus but I'm not sure. But being a medical college and in the path of graduation, it's but obvious that they won't be much promoting into sports and will rather make you focus on your studies, if you have passion towards sports than you can continue it externally from some classes and continue the medical college course as well if you are that very interested in sports. 

Best wishes.

Thank you.