Here is the full list of all 13 Astral Towers achievements.
Heal 20 HP to the Hero during a battle
Win a battle without any spells
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)
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
Defeat the Archmage using just basic tower