I have to create a 4-banner wide roll-up design, but all I have to work with is a 1500px wide image. If I resize this, so it fits the banners, it becomes, logically, very pixelated. I have taken a look at this question, but the result (after importing it in Illustrator) looks quite blurry.
This is why I'm wondering if there is a function or trick in Illustrator to blow up a picture, without it looking like crap. Of course it doesn't have to look like it's not resized, but it should be good enough to take pictures in front of.
The image I'm working with is this one.
There's no magic trick to transform a low resolution photo to a high res image without losing sharpness. If you're willing to let your photo have a more stylized look there are some options. There are some solutions using Photoshop but since you're in Illustrator I would look into the Image Trace function.
I did a quick test by opening up your image in Illustrator and using Image Trace with the 'High Fidelity Photo' settings. This will vectorize your image, allowing you to enlarge it without losing sharpness . But it'll at the same time simplify your image giving it sort of a watercolor feel to it. See example below with your original to the left and the vectorized image to the right.
At a distance it'll look a lot like your photograph but up close you'll see the simplified details: View larger here