Speak to the Catapult Guard. He will tell you about his problem: he fears he's going to be fired, because he failed to keep the catapult in good shape. Tell him that you think he's going to be fired, and he'll ask for your help. He wants you to talk to General Hining, his supervisor, in the Tyras Camp.
Head back into the Tyras Camp, and speak to General Hining. Ask him about the catapult, and he will give you a letter to deliver to the Catapult Engineer. Once you're ready to head out through the traps, leave Tyras Camp again, and follow the map to the Engineer. Remember, each red dot is a trap along the way!
The Catapult Engineer
Speak to the Engineer, and give him his letter. He's also in trouble from his boss, and is more than willing to let you try and help repair the catapult. Ask to read the schematics, and you'll find out you need ten planks, and ninety nails, but you won't know what type. Now, talk to the Engineer again, and ask to read his letter, which will tell you that you need Mahogany planks and Mithril nails, along with a few parts that are supposed to be delivered from the Dwarves.
The Delivered Parts
Head to Ice Mountain, and find Rolad the Dwarf in the Eastern building. He tells you he already sent the parts, but his delivery dwarf has not returned from Port Sarim yet. Make your way to Port Sarim and find Thaki in the bar. Talk to him, and he refuses to hand over the parts, because you're not a sailor.
Go outside, and find the drunken sailor right outside the door. Search him, and take his hat. Put it on, and once again, talk to Thaki. He'll gladly give you the parts, now that you're a sailor. Now, make sure you have everything, and head back to Tyras Camp to fix the catapult!
Fixing the Catapult
Head to Port Sarim, and take the Charter Ship back to Port Tyras. Talk to the Catapult Guard, and tell him that you're ready to build the catapult. Put the pieces together, and then talk to the guard again.
This time, you must aim the catapult. This is actually quite easy, just aim the needle right toward each of the three stones, and experiment with the counterweight to make the shot go for a shorter or longer distance. Once you've hit all three stones, you've completed the quest! Speak to the Catapult Guard again to claim your reward.