Not really a dedicated app to what you are wanting to do, but more like a workaround. You can use a web service called If This Than That (IFTTT). IFTTT basically allows connection between two services (also called channels) by creating a trigger on one service and an action on another.
So in your case, you would create a "New tweet by another user" trigger and a "Post a tweet" action.
Some caveats to using this service; it's limited to specific triggers and actions and doesn't really have much control beyond that. Other than seeing if the user has tweeted, there really isn't much of a way to measure specific activity. You also mentioned in the comments that you only wanted to run this once, which isn't possible in IFTTT unfortunately. Recipes are either on or off.
Steps to creating a recipe:
- If you don't have an account, make one now
- Browse to "My Recipes" and click on the big blue "Create a recipe" button
- Search for the "Twitter" channel and select it (you might have to connect the twitter channel, IFTTT should walk you through this process)
- Select "New tweet by a specific user" and enter in the twitter handler
- Search for the Twitter channel again
- Select "Post a tweet" action and enter in the tweet that you want to tweet (which can include a user mention)