To make it relevant today, you could focus on the themes of self-discovery and empowerment. Let the sleeping beauty have a career or a passion she pursues when she wakes up. And maybe have her break free from traditional expectations placed on women.
One way to modernize it is to change the setting to a contemporary city. Make the curse a result of something like environmental pollution or a corporate conspiracy. Also, give the characters more complex personalities that reflect modern values and struggles.
You could add modern elements like technology or current social issues. Maybe have the princess wake up to a world with smart devices and deal with them. Or have her face challenges related to modern gender roles.