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