If you're running out of room in your backpack, you can place items in the storage Bank in your Dorm Room. Just go to your Dorm and use the Bank chest, move items to and from your backpack to your bank.
If both are getting full, it may be time to sell some items. You can sell most items at the Bazaar in Olde Town, and all items can be sold to regular shopkeepers throughout the Spiral.
If you want the prize, you have to build the ladder rung by rung. There are no shortcuts.