Yes just find an app that will backup your apps and then copy a file with the extention .apk (example.apk) and re download it from Dropbox, find a file viewer application and then use the .apk file to install it.
I believe once you purchased or downloaded an app you can download and run it in multiple Android devices. As long as you use your registered profile. Download the app then sign in with your registered username/email. Not sure if Dropbox works on app transfer.
First you must to enable install app from unknown source. Who to? On your device, goto Settings>Applications>Check the box Unknown sources.
Dropbox is really a versatile app and is a waste if we don't fully utilize it. The method is simple. In your computer, download the apk file to your Dropbox folder. Let it finish syncing. In your phone, open the Dropbox app, navigate to the folder where you keep the apk file, click on it. Dropbox will then download the apk file. The usual installation follows.