Sure, it might cure it, if it's the cause of it. For me, at least, the mattress is the root of almost any back pain I have, which is why I detest hotels with anything softer than solid concrete. I have been told that firmness and back alignment are directly proportional. I was also told that all mattresses are essentially the same once you get below the cushion on top. All the fancy coil designs and spring pockets and other assorted silliness are supposedly pointless. The number of coils does, obviously, affect the mattress, but stay away from gimmicky designs with exotic, expensive hardware. You can usually spot those because they'll have some sort of plexiglass-encapsulated display to show off their neato-spiffy springs.
I do occasionally remind people that your body was made for sleeping on the ground with maybe a pile of straw thrown in for padding. Softer mattresses, the ones with fewer coils or poorly inflated air systems will tend to put a bow in your back. A sailor who does daily stretches and exercises on the deck (and is 19) can usually get away with sleeping in a hammock, but it doesn't do so well for regular folk. Look for something good and firm in the spring department, and then go model by model until you find the top padding that you like. If you go that route, any of the popular brands will be just exactly the same. A friend who worked in furniture liquidation for a couple years reported that many of the big label mattresses come out of the same factory with different labels. There just isn't that much difference.
If you're so inclined, one of these controllable air systems might not be such a bad idea. I've been leaning in that direction the last year or so, since my mattress is almost old enough to vote. If I can get the money together, I might just get one, so I think it's worth looking into to see how it matches your situation.
If your back is really hurting, a doctor visit might be in order. See your doctor, and maybe your podiatrist if you can. Bring in your regular shoes and have him look at them. Many, many foot problems can be diagnosed with relative ease by a good podiatrist taking a look at how you wear your shoe tread. Just as rabbit said, your feet are where it all starts. Quality shoes make a huge difference.
