Demon's Soul's original release is the first of its kind, laying the foundation for later FromSoftware titles like Bloodborne and Dark Souls. Like those games, checkpoints are relegated to specific points on the map. However, unlike the spiritual successors, fewer shortcuts exist to make the paths toward bosses easier once you reach them the first time.
They are still there, but a few of them miss it, making dying a particularly grievous loss. Fortunately, a couple of areas have some hidden shortcuts. While 2-2 has an a shortcut detailed here, 4-1 has an even more well-hidden path to bypassing almost the entire area, particularly the more challenging parts.
The Fourth Archstone
You still have to take on a few enemies and progress a little bit before being able to access this shortcut, but it is mere trifle. You'll only encounter Silver Skeletons and Archers in this opening area, who are pretty easy to best in combat as long as you do not let them gang up on you. There is an opportunity for two or three to attack you at once, so take it slow and try to hit them only one at a time. For mages, fire helps take them out with ease.
Proceed up the stairs and you'll eventually reach a castle gate with two archers on top. Inside the doorway here is a path to the left. Go through here and keep your shield up since you activate a trap which shoots three arrows at you.
Go up the stairs to where the archers are and head to the other staircase. At the top here is where you can bypass the level. You will need a running start and then roll off the edge of the wooden platform over the wall, falling below to where the Regenerator Ring is waiting.
From here it is a simple jaunt to the boss. Follow the path along the side of the cliff, take a left and then a right through the doorway. down the stairs here is the fog door where the Adjudicator is waiting.
The Adjudicator
The Adjudicator is one of the easier bosses, regardless of the build you go for. Mages can beat this monster without ever even going toe to toe and remaining on the high ground.
Melee attackers will have to attack him enough until he falls on the ground, and then attack the bird at the top of his head to actually affect health. A more detailed explanation, along with a strategy detailing all of the other bosses, can be found here.
The area after beating the boss, 4-2, is more challenging, but it also provides one of the most ideal farming locations. Just kill the reaper at the start of the area for an easy dose of several thousand souls. Once you make it through the whole Archstone, an even more ideal farming spot opens up. Use the Storm Ruler to easily take out the Storm Beasts at the final room of the fourth world.