Can old men improve their health with nutrition supplements?

May be. An otherwise healthy person consuming a balanced diet does not need nutritional supplements. If the "old" person in question has difficulty taking usual foods, nutritional supplements may help in maintaining adequate nutrition. Improving health usually requires treatment for an ailment, if any, balanced nutrition and exercise.
Possibly. If they have a identified deficiency, then a supplement may be useful to address quickly. Otherwise, best to stick with whole foods.