Yes, you do have the right to take part in the programme if you have legally resided in Estonia for a period shorter than five years.
You have to apply for with the Police and Border Guard Board to participate in the Settle in Estonia Programme. You can file your application here, send it by e-mail (firstname.lastname@example.org) or post (to the Estonian Police and Border Guard Board, Pärnu mnt. 139, 15060, Tallinn) or do it in person in one of the Police and Border Guard Board’s service points. The locations of the service points are specified here.