How can I achieve the noise that this video has?

I'll assume you're talking about video noise ... "grain" ... and not audio noise.

And that's ... the 65 gazillion dollar question. I've seen all sorts of things listed as THE way to emulate film grain.

In Premiere Pro, there is the Noise effect in the Video Effects/Noise & Grain bin. Not a lot of options, but it's mostly ok.

In the Obsolete folder are a couple others (that will at some point disappear) ... Noise HLS is my fav of the two, you have more controls. Past that, there are a number of noise generators from Maxon/RedGiant through several others.

Some go way out there ... as the original noise was film grain. The size of the grain varied by 'density' of exposure. So shadows and highlights had slightly different sizes and amounts of grain. There are YouTubes out there on how to create those that can be extremely ... dense.

Walter Volpatto, one of the most noted colorists currently working, no longer does the amazing grain parade creation. As in his testing, he produced a typical deliverable file for theater, broadcast, streaming and web use. And then viewed them as it will be distributed and viewed. In a projected theater, streamed & such. And found he couldn't tell any details of how he'd created the grain.

All distribution methods involve a couple different steps of compression and transcoding, and the end visual result for the viewer ... was little dark flecks. Period. You have, essentially control over how many and how big, sort of. But realistically, not much.

So he just does a basic 'grain' setting with the vieweing environment in his grading suite set up for the expected viewing situation. If it's visable but not a problem, done.

Seems simple enough.

Neil