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.
Pass through the Al-Kharid/Lumbridge gate without paying
Visit Death's Domain
Equip Fancy Boots or Fighting Boots from Stronghold of Security
Use the Home Teleport Spell with any spellbook
Equip a Mithril Scimitar
Vote in a Poll or view Poll results
Complete a Clue Scroll
Complete the Demon Slayer Quest
Complete the Tutorial
Complete the Vampyre Slayer quest
Reach a total level of 100
Enter the Wilderness
Complete the Cook's Assistant quest
Complete the Dragon Slayer quest
Reach Combat level 20
Ride the canoe in Lumbridge
Chat clothing with Thessalia in Varrock
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)