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