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