So you have custom Java code that doesn't use a test framework? You should ensure your code can compile into a JAR file that is command line executable. Otherwise, the alternative approach is Java (servlet) code that runs on web server, which depending on what you do, can be a trivial or daunting task.
You could craft a simple web UI that handles file uploads and invokes a shell execute against the JAR with the uploaded spreadsheet(s) as arguments. This you could do in any choice of language. You can track progress or manage the execution by perhaps forking off the execute so it's not blocking to the web request/server, and log down the process ID of the running JAR for checking status or killing it if needed when hung. Other status could be logging/redirecting the JAR's stdout to file and simply be able to present that file data in web UI for status progression.
Rewriting everything including that Java code logic is likely a lot of work unless your program is simple. Best to either adapt the Java code for Java web framework executability or integrate it with another web framework using shell execute of the JAR.