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