Hi
You can find the blur tools/effects from the overhead menu Effect > Blur (under Photoshop Effects) and choose one of the effects to blur your image.
Illustrator, as a vector-based software, doesn’t have a realistic Blur Tool. So if you want to blur an image I would recommend you to use photoshop, which will give you more precise effects. In Illustrator you can blur edges of images or shapes.
To blur an image in Adobe Illustrator you have to simply select the image, and choose a blur effect in menu Effect > Blur (under Photoshop Effects)
There is also another option to blur under Effect > Stylize (under Illustrator Effects) > Feather.
In order to add grain
You have to duplicate the shape or shapes that you want to add grain and go to Effect > Texture > Grain
And here you can experiment with the intensity, the contrast and grain type (I would recommend you to use Stippled format).
Once you’ll finish and set the desired type of grain click ok. And open window / transparency Here you’ll have to play with the transparency methods like overly or multiply to add the grainy effect transparency.