Answer
Employers are not required to have written policies, and they are not required to offer paid time off at all.
Just because a policy isn't written down doesn't mean it isn't enforceable. And, obviously, since you know about this particular "policy," it is well communicated.
Your options are:
1. Work because your day off will not be approved.
2. Take the day off without permission and subject yourself to the employer's disciplinary action, which could very possibly be termination of your employment.
3. Quit and get another job with time off policies more to your liking.
Employers are not required to provide paid time off, and employers are allowed to set the work schedules and work rules for their staff, so if your employer doesn't want to allow you time off, they don't have to. They don't even have to approve requests for time off if you have paid time to take.