Running is a great option in terms of a cardio workout to help increase your lung capacity. Any activity that you're doing aside from your normal daily activity, walking around, maybe your job, running increases your heart rate, increases the capacity that your body requires for you to get oxygen, and will naturally over time, increase your body's lung capacity.
This does take time however. You do want to wait a few weeks. Any adaptions that you're waiting for your body to go through will take about four to six weeks. So if you are using running to increase your lung capacity, have patience with it. Any adaptations you're waiting for your body to go through, will take about four to six weeks. So do those two, three, four times a week runs and over time, you'll start to notice a difference in your lung capacity.