Well, in my case, I'd have to find out how much insulation is in the wall first; if it's lacking insulation, fill that in.
Then double up the drywall with green glue (simplest). Would be smart in the long run to decouple the wall also.
http://www.tmsoundproofing.com/decoupling-explained.html
On the matter of noise coming from the 'new' neighbour,
There is this soft rattling / humming noise coming from their side. It sounds like a machine, maybe a heater, humidifier of some sort.
I can only hear this in my bedroom which has the only adjacent wall (other than bathroom) with them.
I've been living here close to 9 years now and never heard this before. Funny thing is, I've only heard this 3 times so far, each time right after I made noise complaint against them, late at night. Coincidence?
Question is, how do I go about finding out where/what is causing this noise? It would be simple if I could go into their unit when the noise starts.
I'm trying to get a floor plan of their unit from management but it's been slow process.
Currently all I can do is, when the noise starts again, have the concierge listen to the noise from my room and convince him to talk to the neighbour to find out what's causing it.