Though Arcadia was designed as a two-player game, the official rules of Arcadia include rules to play the game on your own. You can read those here. Though these rules outline the basics of solo play, there are some difficulties with them—most notably, it is impossible not to win with these rules! To resolve some of these issues, have a look at Jackalwere’s suggestions.
Keep in mind that several of the other variants can be used in solo games of Arcadia as well.
The Legend of Arcadia
The most developed variant for single player games is perhaps The Legend of Arcadia, a modification of Arcadia’s single player rules designed by Jackalwere that follows the Quests of both The Wyld Hunt and King Ironheart’s Madness in order,and uses the entire map of Ardenmore and Middlemarch. It thus tells the entire story of Arcadia, and your Character grows as it continues on its Quests. Though it is best played with a complete set of cards, it can be played if you are missing some cards, by substituting missing Quests and Leagues with proxy cards (using these detailed descriptions of both card sets). If you want to play this, make sure to read this update too. Examples of solo Quests are also provided.
Into the Unknown
In this variant, you build a map of Arcadia while you travel through its Leagues, rather than doing so at the beginning of the game. Your Character has not seen much of Arcadia, and is discovering its lands as she travels through it to complete her Quest.