Colombo Archery Centre (Longdon Place) runs training courses. Call them and find out how to join.
https://archery.lk/
If you want to try first, visit Laya Resort - Kukuleganga. They have archery facilities (you pay extra) and where you can train with an instructor.
It's not easy as you think, as it needs quite a bit of strength & steadiness in your arms. But they have various bows that carry different tensions.
You can spend the day there and they have a day package that includes lunch & tea. You can stay overnight too. There are nearby places to visit like the Fa-Hien caves, Makeliya Ella etc