Curtains for a non-air-conditioned apartment in the tropics
I grew up in the 1940s and 1950s in hot, humid south Louisiana and hot, less humid Dallas, Texas. We had no a/c in Louisiana and starting in about 1954 in Dallas we had window units in some rooms, but not in the bedroom I shared with my brother. In 1960 we moved to a new house in Dallas, Texas with central a/c. The improvement in comfort was astounding.

After leaving home I sometimes lived in downscale student digs without a/c and it was sub-optimal, but obviously tolerable. One place was cooled with an "evaporative cooler" or water fan. Mumbai is presumably too humid for those to be effective. For two years in my mid twenties I lived comfortably without a/c in Fiji near Suva, but Fiji's summer high temperatures are low compared to Dallas or Mumbai.     

If you can afford it, I do not think you should deny yourself a/c in Mumbai on the grounds of damage to the environment.

But as far as other interventions on your existing house, outside louvers over the windows would be better than curtains inside. Louvers would reduce solar gain while still allowing airflow into the room. The sun will heat the louvers but much of the heat will stay outside.