Find the Elixir of Spring Prime Recipe (gatherable at ( -52, 207, 6 ) /waypoint -52, 207, 6, looks like a sparkling piece of paper). Scribe this for later use.
Kill all five of the non-aggro roaming training soldiers. This causes Sergeant Schriber to appear briefly, and he unlocks the doors to the other areas. [This is not a required step in the quest, but is needed so as to be able to continue with the quest steps.]
Gather the following ingredients:
Three sulfur rocks. These are harvestable in the barracks area, around the edges of the room centred at around ( -60, 210, -30 ) /waypoint -60, 210, -30. They look like grey stones against a grey floor, so they can be difficult to see.
Once the above ingredients have been acquired, craft the Elixir of Spring Prime at the Bubbling Cauldron in the outer kitchen area at around ( 65, 210, 52 ) /waypoint 65, 210, 52. This is very easy, no tradeskill levels are required.
Return the elixir to Vethilot the Young to complete the quest.