Yes, you do not need to have any existing Hong Kong bank account with another bank to apply for Neat Business.

Moreover, after you have opened your Neat Business account, you do not need another bank account in order to operate your business and you can use your Neat Business account for any official purposes. Neat statements can be used for, for instance,  audit, visa application, or linking your account to third party services.  

There are some differences between a bank account and a Neat Business account, you can find them here