Bring the Piper Cub to a certified aircraft mechanic for an FAA inspection and STC approval. Once approved, make the appointment for the installation of the new Lycoming IO-390 engine.
Take the Cub to the mechanic for the installation and bring all Cub engine and airframe logbooks and maintenance records. The mechanics perform the engine installations and will note the changes, along with the FAA STC into the logbooks. Once the engine is installed, you must keep the STC in the maintenance books and records for the duration if the aircraft's life.
Take to the air for a test flight. Use the throttle control to rev the new engine and increase the Cub's speed. Take two to three training hours to familiarize yourself with the enhanced engine performance and nuances it gives the aircraft.