How can I hire a professor to answer my advanced technical questions in a niche area?
Unless you are in a place where professors are greatly underpaid and generally abused, this seems like a quest that is bound to fail. People are busy and they aren't normally driven by money (other than large pots of grant money). 

What I think you want, instead, is a collaboration. That can be started with just an email, provided that you have something to contribute yourself. This requires you to do a bit of research to find someone with common interests to form a two-way partnership. 

Perhaps the story of Srinivasa Ramanujan and his relationship with Hardy will be instructive.