In 2013 he became a European Junior champion. At the 2013 FINA World Junior Swimming Championships in Dubai, United Arab Emirates, Rapšys competed in 4×100 metre mixed medley with Rūta Meilutytė, Povilas Strazdas, Eva Gliožerytė. During 4×100 metre mixed medley heats, they finished third with a time of 3:55.74 seconds and qualified to the final. At the 4×100 metre mixed medley final, Meilutyte, Rapšys, Strazdas and Gliožerytė finished second and won silver medal with a time of 3:52.52 seconds. In the 2013 World Aquatics Championships he reached semifinals in the 200m backstroke swimming.