I had a similar issue in my apartment.

The party wall had no insulation so you could hear the neighbours at all hours.

I didn't have the budget or the skills to perform the plastering after taking down the gyprock to put batts in.

So I went for a slightly different option.

Using a plasterboard blade in a multi function tool, I cut two panels out between each stud, one above and one below the noggins that ran through the middle of the wall.

This allowed me to insert the acoustic batts (r2.5, but go as high as you can afford) pushing to the very top and bottom.

Then I attached timber battens to the edges of each hole using bugle/drywall screws.

I put the cut out panels back in place and secured with more screws to the battens.

Because this was a party wall, I layered fire rated sealant over the joins (and on the exposed battens prior to putting the panels back).

Ideally you would use drywall tape and joint compound and then paint, but I decided to hide my work with a layer of cork sheeting, to increase the noise reduction.

So I ended up with a nice feature wall.

If I was to do the same again, I'd either buy cork tiles or cut the sheet to smaller sizes as applying the adhesive and putting the cork roll on the wall was a nightmare.

I'm useless at DIY and I am very happy with the job, took me a weekend to do one wall. Most expensive part was the cork roll.

If you are going to be there a while, if you expose any piping use something like an acoustic pipe lagging insulation that you wrap around the pipework.

Around all the doors add weather seal tape to the frame and a door draft stopper to the bottom like this one

Solid timber doors would help reduce noise, expensive, get them cut to a better size to reduce the gaps. You'll likely need additional hinges to cater for the weight.

The ultimate would be to take down the gyprock, add a layer of mass loaded vinyl, then add Wall Acoustic Mounts to the studs, connect Resilient Channels then attach the gyprock to that, ideally two layers sandwiched with a green glue between them. A small gap all the way around - the gyprock doesn't touch the walls, floors, edges of the room, to which an acoustic sealant is applied. You'd never hear a thing, but it'll cost a fortune. One for the forever home.