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