So what are the benefits of zinc, and can you buy zinc tablets in a bottle?

I keep hearing about it in the past week. I know it's beneficial. Head and Shoulders has zinc in it to help get rid of dandruff. If regularly using zinc on sun-damaged skin, will it help a lot?

Also, what is the benefit of zinc oxide ointment? Is it just as beneficial? Would taking a zinc tablet once or twice a day be better than applying ointment?

Zinc, most of the time prevents you from getting sick. But you should only take it for a week at a time. This is not something that you should constantly take. I don't know what will happen if you consume to much.

Zinc: What is it?
Zinc is an essential mineral that is found in almost every cell. It stimulates the activity of approximately 100 enzymes, which are substances that promote biochemical reactions in your body. Zinc supports a healthy immune system, is needed for wound healing, helps maintain your sense of taste and smell, and is needed for DNA synthesis. Zinc also supports normal growth and development during pregnancy, childhood, and adolescence.
Amounts of zinc provided in these studies ranged from 4 mg a day up to 40 mg per day and were provided in a variety of forms (zinc acetate, zinc gluconate, or zinc sulfate). Zinc supplements are often given to help heal skin ulcers or bed sores.

I, personally, have found that taking zinc supplements helps to improve brain function and memory. It also helps with healing any skin irritation, acne, lesions, etc.
Taking zinc supplements and applying zinc ointment to damaged skin would give you added benefits.