A few kam jugadu tips which comes to mind -
1. If the door is made of hard wood/metal, get it replaced with 1-1.5" thick ply door with good fitting(not more then .5 cm space at the bottom). should cost around 1.5k.. Additionally you could fit the long rubber strip at the bottom of the door, which touches the ground(you can get it for 20-30 bucks from any leather workshop). We did this on the main doors to keep out snakes etc
, wandering off from the garden. If the chokhat(the outer frame which holds the door) has bottom support as well like windows, which doesn't leave any gap then much better.
2. If not going to use the window, then close the window, try to fill the gaps with newspaper/polythene any useless stuff which can be stuffed and fit a large cardboard(single piece) perfectly on the outer windows(from inside). I see that you have good enough budget then you can try double glass windows.
3. Generally sound doesn't pass through 10" thick walls, but older apartments usually have 6" thick walls made of traditional bricks, if so then cover the outer walls(from inside) with the 1/2" plybords and paint it with good colours. Shouldn't cost more then 10k for one side.
EDIT: For me, cheap AC's noise is very irritating when no other sound is coming from outside
. I mean may have to invest in the Good ACs