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)
Craft some Air Runes at the Air Altar
Equip a Mithril Scimitar
Catch a Tuna at Musa Point
Visit Death's Domain
Chat clothing with Thessalia in Varrock
Pass through the Al-Kharid/Lumbridge gate without paying
Use the Home Teleport Spell with any spellbook
Equip Fancy Boots or Fighting Boots from Stronghold of Security
Vote in a Poll or view Poll results
Ride the canoe in Lumbridge
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