Anything that's old-hat has grown so familiar that it's boring or seems old-fashioned. Your grandfather's war stories might be fascinating to your friends, but after hearing them so many times, they're old-hat to you.
A lot of what your'e studying in school might be exciting in September but completely old-hat by May. The same could be said of your lines in a play: they're a challenge to memorize and then thrilling to say on opening night, but if the show runs for a month, your part will soon be old-hat and you'll be ready to audition for a new role. This adjective was first recorded in the early twentieth century.