I'm planning to visit haikou during this coming December.

Only one bus is scheduled daily at Wenchang Bus Station for Wuzhishan City. It takes 4-5 hours and costs CNY48 per person. In the next morning, at the drop-off long-distance bus station, you can take a direct bus to the foot of the mountain (Shuimanxiang). I remember the first bus leaves at 09:30. It takes an hour and costs CNY10 to reach.

It is safe enough for you two to climb, and there is no rule about foreigner not being allowed in without a guide. Don't worry! As the mountain is quite popular, you can follow a group or other visitors to climb so that you won't get lost.

Have a good trip!

Hello Chee. For the transportation, I have another suggestion:
you can take a bullet train to Haikou East Station first within 30-40min. Take a taxi to local South Bus Station within several minutes. There you can easily take a bus to Wuzhishan City, for which buses are scheduled from 07:15 to 18:30. Then head to Shuimanxiang to visit the mount.