How do you get the Season 2018 - Solo/Duo Gold Icon?
It was a ranked reward. Riot's description in the client reads: “Earned by achieving Gold in the 2018 ranked season in Solo/Duo.”
Is the Season 2018 - Solo/Duo Gold Icon still available?
No. The Season 2018 - Solo/Duo Gold Icon was a ranked reward: to get it you had to reach Gold in Solo/Duo during Season 8 (2018). Riot does not give out past ranked rewards again, so it cannot be earned anymore.
