You can definitely sand it down, depending how far off the surface it broke that might take quite a while as ceramic is a really tough material. Then again that should take off the anxiety to destroy anything in that regard. With enough care even a grinder can be used to aid you in this process.
Depending on the desired visibility of the repair you can then paint over with good quality acrylic paint matchinf colours or going the extra mile to take your chances to append on the original design of the cup with some fun little motive, extending the personal story and connection to the piece in the philosophy of kirigami.
However you might decide to do this, I would be very happy and interested to see your results here on the magazine in a follow up post. Wish you all the best with your project.

Disposable nail file and a couple minutes of elbow grease.

Or the course unfinished ceramic edge on the bottom of another ceramicware.