The General Hydroponic kits you are speaking of DO indeed run off of an air pump. They pump air below the water level into the feed column, forcing air and water up into the drip ring and onto your media.

You can indeed use one air pump to operate the drip ring and to aerate your nutrient solution. I am not familiar with the pump that comes with the kit, but make sure that it moves enough air volume to both operate the drip ring and sufficiently aerate the nutrient solution.

There is a grow in these forums which is stickified (is that even a word?). Here is a link: http://boards.cannabis.com/grow-log/...-dwc-grow.html

That grow uses something similar to what I think you are wanting to do, so check it out.

I'll say also that in my opinion, you really don't need the drip ring. As long as you raise your water level to reach the bottom of your netpot, you're good. After roots begin to grow lower than the netpot, you can lower your water to about an inch below the pot.