How do you get the Season 1 Reward - Bronze?
It was a ranked reward. Riot's description in the client reads: “Earned by achieving Bronze in the 2011 ranked season.”
Is the Season 1 Reward - Bronze still available?
No. The Season 1 Reward - Bronze was a ranked reward. To earn it, you had to reach Bronze during Season 1 (2011). Riot does not hand out past ranked rewards again, so you cannot earn this icon anymore.
