The best joint to use is pork shoulder as it has the right amount of fat to keep it moist during long smokes and allows it to pull well. Even though it is called "Butt" it does not come from the rear end of the animal. The word Butt comes from the wooden casks in which they used to be packed in Boston - yes like "water butt".
You can buy them either bone in or boned and rolled. If rolled then snip the string and flatten it out before smoking. This will give you more surface for the rub and also a more even cook.
Ade - Here is one of the boned pork shoulders I have bought for this weekends briquette comparison. It weighs just over 6 Kg (13.5 lbs). You can see how much fat marbling it has - which is what makes it so good for pulled pork. Butchers will usually sell these rolled and tied however I prefer to buy them flat. You would be surprised what some butchers try to hide in the middle when it is rolled - it is a good place for some to hide a lot of additional fat. Even if you do buy it rolled It is best to cut the strings and open it up before using.
A lump of shoulder this size will fit in the Weber (just) however I usually cut them in half - but you could cut them into smaller chunks which will make them more manageable to smoke.
This is now ready to be vac packed and placed in the fridge for 36 hours. 36 hours isn't a magic time - purely convenience. I just find that I usually pick the meat up in the evening on my way home from work and I start cooking early in the morning. If I bought it the day before and applied the rub that night it would only get about 12 hours. I therefore usually buy it 2 days in advance which which means that the rub gets a day and a half contact in the fridge.