You could use the Dropwizard framework
Dropwizard provides a complete web application environment, including servlet container, which is deployed as a single executable JAR file.
For example you could deploy a Dropwizard application to, say, Heroku or Amazon EC2 and have the application respond directly to web requests. Here is a tutorial. This is probably a lot simpler to get running than the complexity inherent in JEE applications with all their configuration.
How console and web applications are different
As @MichaelT has mentioned in his article you will need to make some modifications to your code to make it a web application and this is not trivial. Almost without exception, console applications retain state between input and output from the user. Web applications do not so you are fundamentally changing the architecture of your code.
Think of it like having a single console running your application and many users all sharing the keyboard (they can take turns but you can't predict who's turn it is). Somehow you have to ensure that user1 is only able to mark exams, but user2 can only create exams and so on. Essentially after every input to your console app the screen resets back to the start so any user can step up and start work on the keyboard. Clearly the first entry they make must be an identifier so that user1 can jump quickly to where they were a moment ago before user2 got their hands on the keyboard. This is the role of session IDs in web applications, and they typically get handled automatically by the browser in a cookie.
Introduce layers to expose common code
To modify your existing application to turn it from a console app into a web app you need to split your console app into some layers that represent shared code. For example, once you have a user identified you will call into some data retrieval logic that connects to a database and gets the exam that they will work on. This code would be common to both the console and the web applications, so could be moved into a shared JAR.
Eventually you'll find that you have code that just handles the user interactivity (the UI layer), another JAR that handles special processing logic such as data validation and collating exam results (the business layer) and yet another JAR that just looks after data persistence and retrieval (the data layer).
Create the web-specific layer
Once you reach this point, you can create a specific web UI (the wrapper that your friend is talking about) and hook it up to the business and data layers JARs just by including them in your build. The web layer will contain views that output HTML for rendering by a browser, and it will have endpoints that will retrieve exam data in response to a GET request or add new exam data in response to a POST request.
Overall, this would be a good exercise for your personal development as a programmer so definitely persevere with it.