How can I add grain or noise to a picture like this one?

Here’s a fully non-destructive way to create something similar using Photoshop. The beauty of this approach is that you can adjust the filters (or even remove them) if you don’t like the result, without doing the whole edit again.

1. Right click the image layer, and choose Convert to Smart Object

2. Click Filter > Camera RAW filter.

3. Click on the fx tab in the Camera RAW filter.

4. Move the grain amount, size and roughness sliders until you get the effect you want.

You could of course stop right here and you’re done, or go on to attempt to colourise the image like your example.

5. In the Camera RAW filter click on the Split Toning tab, then you can apply toning to the highlights and shadows to make it simililar to your example

I continued to mess around a bit more, adding a Curves (point) adjustment in Adobe Camera RAW, and then tweaking the Split Toning again to get lighter more bluish/greenish shadows.