How do I have to process an image to test it in a CNN?
MNIST is an easy dataset. Your model (CNN) structure may do quite well for MNIST, but there is no guarantee that it does well for more complex images too. You can add some more layers and check different activation functions (like Relu, Elu, etc.). Normalizing your image pixel values for small values like between -1 and 1 may help too.
I have trained my CNN in Tensorflow using MNIST data set; when I tested it, it worked very well using the test data. Even, to prove my model in a better way, I made another set taking images from train and test set randomly. All the images that I took from those set, at the same time, I deleted and I didn't give them to my model. It worked very well too, but with a dowloaded image from Google, it doesn't classify well, so my question is: should I have to apply any filter to that image before I give it to the prediction part? 
I resized the image and converted it to gray scale before.