Push isn't incorrect but it does have a certain sense to it. I'd write the sentence as you have it if I wanted to give readers the idea that the person I was describing was particularly uncouth. If I didn't want to give this idea, I'd say he raised a tissue to his mouth.

Usually a person pushes something that is heavy, such as a broken car, a box, etc. Pushing requires effort. In your example, an old man making such an effort to push a tissue illustrates how weak and feeble he is. Try to visualize the old man raising his arm very slowly, as if the tissue weighed a lot, and it might seem that he was "pushing" the tissue.