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