is this sort of approach normal?
No, it is not normal. It doesn't happen much often. Generally, publishers do thissend these kind of Inmails, while searching for authors. But, I haven't seen a mail like this for devs.
However, startups generally look for people to help them in their projects. And as they cannot afford full-time devs, they generally look for freelancers or passionate people who might love to do it as a side-project. So, please confirm the same from the person.
How do contracts like this work? Hourly, by the project?
These type of gigs are no different from a freelancing gig. So, you would be paid hourly(unless informed otherwise).
Recruiters do tend to use those sugar-coated words. So, you don't need to get all skeptical about that. Just reply back to her asking about the details of the company, the project and how the contract would go about.