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