Email the contact academic and explain your academic background, and say something like 'Is this the sort of background you are looking for?'. This is the ice-breaker. If they like what you have told them, you should get a positive email back, and encouragement to apply.
Any application needs to be tailored entirely to that project, as stated in the ad. Other than that its essentially the same process as any other PhD application - why this project interests me, and this is how my academic background makes me perfect for this project.
PS. Far fewer people apply for specified project titles than 'open calls'. You chances of getting an interview therefore are much higher.