good budget hotels in Haikou city centre near bus stops

Yes, in general, you will find several budget hotels in Longhua District.
Hanting Express (Haikou Guomao Hot Spring) sounds like a suitable one. The price is CNY 100 to 150 per night with comfortable living environment and warm service. And the surrounding transportation is convenient. The address is No. 51, North Longkun Road.

Is there a direct bus from the HK Railway Station to Hanking Express? What is the number?

Yes, you can take public bus no. 28, get off at Affiliated Hospital of Provincial Medical College.
After arriving, walk southwest along Longhua Road and walk southeast along West Yilong Road to arrive at the hotel. The total walking distance is about 630 yards.