Nsgach - sorry, just providing ideas. If you do it your self, you can do these projects pretty cheaply. But I do agree that double walls and the like can be expensive if you have to pay someone to do it. In fact, I said that a double wall was not inexpensive. But also, this is not a very big wall.
My first recommendation was dry wall with green glue. Did you read the website I referenced? One level of drywall gives a STC of 40, adding a second level goes to 42, adding a second level with Green Glue goes to 52. Using Green Glue with drywall will do a lot more than just adding drywall. Green Glue is not cheap, but it really is effective. Mass is great, but isolating the surfaces is very efficient. Green Glue does a good job of that, by essentially turning the sound into heat, as I understand it. If you have not used it, you should look into it.
1 Layer of Drywall STC 40
2 Layers of Drywall STC 42
2 Layers of Drywall with Green Glue STC 52
My first recommendation was dry wall with green glue. Did you read the website I referenced? One level of drywall gives a STC of 40, adding a second level goes to 42, adding a second level with Green Glue goes to 52. Using Green Glue with drywall will do a lot more than just adding drywall. Green Glue is not cheap, but it really is effective. Mass is great, but isolating the surfaces is very efficient. Green Glue does a good job of that, by essentially turning the sound into heat, as I understand it. If you have not used it, you should look into it.
1 Layer of Drywall STC 40
2 Layers of Drywall STC 42
2 Layers of Drywall with Green Glue STC 52