Here is the full list of all 20 Old School RuneScape achievements. It takes around 0-1 hour to unlock all of the achievements on Windows.
Complete the Dragon Slayer quest
Complete a Clue Scroll
Catch a Tuna at Musa Point
Pass through the Al-Kharid/Lumbridge gate without paying
Visit the General Stores in Lumbridge, Falador, Varrock, Edgeville, Rimmington, Al-kharid and Karamja (Musa point)
Craft some Air Runes at the Air Altar
Equip a Mithril Scimitar
Visit Death's Domain
Chat clothing with Thessalia in Varrock
Equip Fancy Boots or Fighting Boots from Stronghold of Security
Enter the Wilderness
Ride the canoe in Lumbridge
Use the Home Teleport Spell with any spellbook
Vote in a Poll or view Poll results
Complete the Tutorial
Complete the Vampyre Slayer quest
Complete the Demon Slayer Quest
Reach a total level of 100
Complete the Cook's Assistant quest
Reach Combat level 20