Why does everyone forget this other method.
You do not just need to use iTunes any longer.
You can simply tap on an app in the App store you want to download.
If there is an older version of the App, left by the developer on Apple's servers, you will get a pop-up window stating that there is an older version of the app for your device. Would you like to download it now?
Tap OK and the App Store should download the older version to your device.
If there is no older version available, then there is no version of the app for your device and no pop-up window/panel will appear.
Good Luck!