Im one of those once a month or so controllers >_< Kinda busy right now but I’ve been able to be part of the team despite that. New procedures are often tagged in chat so we can follow up.
But replying to your post, personally, I would spend close to the entire time watching the screen, at times I refresh the incoming traffic page to look for inbounds and see if I can attend to them. (Eg. incoming in 50mins, I intend to stay for 45mins but I have time, I would extend a little to provide that service whenever possible.)
This is because pilots can spawn in any time and some could be making trouble (though a rare occurrence). Whenever I’m controlling, I just pretend like its a job, so gotta pay attention all the time xD