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