No, your Estonian company will not expire and you won't need to update it. Even if you don't have any business activity, the company won't expire. If you wish, you can always close down your company or hibernate it.
Please note that by law an Estonian company cannot be 'paused' or 'hibernated'. It can be either active, in dissolution, or permanently closed. However, for Xolo Leap customers we can offer you a chance to downgrade your Xolo Leap service plan to 'hibernate' status.