It appears that you have a "new construction" fixture. These are designed to be installed and accessed when the drywall is down. So to work on this fixture from below, you'll have to cut an opening in the drywall. I would suggest making the opening smaller than 2'x2' so you can use the drywall patches sold by most home improvement stores. Make sure two sides of your cut are in the middle of the joist so you have something to reattach the drywall to when you're finished. And be careful not to cut too deep when opening the drywall, since there could be electric and other utility lines very close to the back side of the drywall.
Rather than just replacing the ballast, you might want to consider replacing the entire fixture with a retrofit or "old work" fixture that's designed to be installed and accessed after the drywall has been put up. The fixture is attached to the drywall directly, instead of to the joists. And the electrical is designed to be pulled through the opening when the fixture is dropped from the ceiling.