The answer to your question will depend in large part on how you are using UIApp. In most cases it's not as simple as changing a few lines of code but more of a script redesign.
If you are using it to create a custom user interface, your path forward would be to rebuild your interface as HTML/Javascript and serve it to the user with HTMLService. The service is overall much more powerful/flexible and provides a better end-user experience but it means rewriting how your script works. If you want to take a look at a helpful example on how the HTML Service works take a look at the HTML Service Guide or this quick-start example of an add on.
You can still link this html interface with your script using the google.script.run service but you will need to redesign your code/application to work with the new input methods. The good news is although UI service will sunset on June 30, 2015, "Service will no longer appear in autocomplete, although existing scripts should still function."
It's worth considering alternate ways to accomplish the same goal. The best way to do that is to start by asking "What is the intended use of this script? What problem does it solve?" There may be a better way to accomplish that task which may not even involve updating the script.
