Here is the full list of all 13 Astral Towers achievements.
Reach Damavik town
Complete the campaign at the "Normal" difficulty level
Complete the campaign at the "Hard" difficulty level
Defeat a Master Mage AI opponent
Defeat an Archmage AI opponent
Collect all the skills during the campaign
Get 5 undead creatures on the field (and win)
Heal 20 HP to the Hero during a battle
Get 5 buildings on the field (and win)
Get 4 tower upgrades (and win)
Deal 30 damage to the opponent's tower during one turn
Win a battle without any spells
Defeat the Archmage using just basic tower