The next destination is the dungeon entrance in the lower right corner of the fortress, so you basically need to make your way down. However, before you begin your descent, you should pay a visit to the Gunner to the left side of the gap that lead you to the second dungeon entrance. In here, you'll meet the Gunner.
When you enter the Gunner's chamber, the room will be dark, so you will use up a lamp. Additionally, the room contains a good number of blue blocks, but if you've collected a power-up item for the Fighter, you can clear some of them out by punching them if you think you need to space. Fighting the Gunner is a lot like fighting a Blaster. Get inside quick, duck low, and punch as frequently as possible to knock him out fast. When you do, you will possess the Gunner's body and have 3 units of health to start, so be very careful.
Drop down to where a Blaster waits below, and drop into the pit next to it. When you land, you will be in very tight quarters with a caterpillar. You may inevitably take some damage unless you successfully guard by pushing backwards, but if you stick to the left side and duck down as soon as you land, you should be able to destroy it quickly. Then shoot a tunnel through the wall on the right to continue on.
Shortly after a ladder (which travels up to a room where you can collect energy and reinforced bullets), you will see a pit and two bats. It is important to make it over the pit to the other side. However, the low hanging ceiling may make the jump difficult. You must successfully pull off a double jump here by shooting between jumps and before you fall down the pit. It is nearly impossible to avoid getting struck by the first bat when you land, but make sure you don't get pushed back into the pit by it. Stand over the false floor to the right to proceed down the fortress.
When you land below, you will see a Mummy on your left and a ladder going down on your right. Jump down the ladder, and you will land in the room with the third dungeon entrance. As usual, a Bouncer is in the room waiting to deal with you. With the Gunner's weapon, you should have no problem taking it down before your approach the dungeon entrance.
This dungeon is quite a bit longer than the previous two. There are also sections of darkness, but there are as many as five lamps to collect on your way through, so you may actually end up coming out ahead on lamps in the end. Remember that your bullets pass through walls, so you can shoot bats that are on the other side of low hanging ceilings in order to remove them before you crawl beneath them.
When jumping down off the right edge of the platforms to the floor below, watch out for the Skeleton that roams around on the third level down. If you face right, jump off the ledge, and push left as you fall, you should land sufficiently far enough from the Skeleton to get a good shot off before it gets too close to you and attacks.
The boss of this dungeon isn't particularly strong, but it attacks by dripping fluid on you. If you stand in the ditch to the right of the area with thee statues, you can remain there relatively safely, and jump up to shoot at the boss in order to defeat it. You should be able to defeat it in three or four shots. Then run to the right and destroy the last statue to collect the LARNUA idol.
When climbing back up the ladder on your way back to the fortress, be careful of the first rock enemy you encounter. As you leap off the ladder, you may need to fire at the rock, and jump again to avoid falling back in the hole. Every time you leave and return here, you will be using up a lamp, so be very careful.
Take note that there is a bug in the game which can cause a hole to appear in part of the floor if you destroy a statue while squatting down. If this happens, do not try to collect the item which appears and avoid falling down the hole. Nothing worse will happen other than the need to retrace your steps.
Again, the trickiest part of your return will be the right edges of the dungeon where you must leap up the platform above. Bats will be a particularly tricky obstacle to overcome, but the rocks which fall from the ceiling are not as difficult as they appear. They have a range of movement that you can observe and use to your advantage as you climb back up to the top. Watch out for the Bouncer when you return to the fortress.