I’m not a JapanDoper, but here’s my advice! (I studied abroad in Japan for a few months this year).
Youth hostels are perfectly doable in Japan. I stayed at one in Kyoto; it cost 2300 yen (about $21 per night for a bed in a 4 person room. It was small–smaller than any youth hostel I’ve ever stayed in (there was only like 2 showers for everyone)–but clean, pleasant, and convenient. I can’t give you advice about specific hostels in Yokohama, but do keep in mind they sometimes have curfews, unlike in many other countries. So if you’re planning to be out late check that out, and always read reviews from other users, etc. If you don’t do a youth hostel it can, of course, be crazy-expensive. However you can sometimes can a place in a cheap-ass ryokan (actually a minshukan–ryokan are priiiiicey) which are like cheap traditional Japanese guesthouses. I stayed in one in Kyoto as well; it was about 5000 yen per night. Considerably more than a youth hostel, but you get the whole tatami mat/futon experience. If you want that, of course! There may be some food preparing facilities at either of these places, but I would recommend eating at convenience stores and ramen/dollar sushi places. They have really quite tasty bento there, and for a cheap snack I strongly recommend nikumon, which are delicious (a kind of steamed dumpling with unidentifiable meat in it. You can just ask the cashier for nikumon.) It’s filling and only costs about 100 yen. Also, eating out is really almost as cheap as buying groceries because they are very expensive. A small 1-serving pack of chicken can cost around 200 yen; an apple costs 100 yen if you’re lucky and go to a 100 yen shop. Rice is also very expensive and mostly only comes in big honking bags that cost like $20 bucks. Definitely go to these if you do shop, by the way–it’s really not as sketchy as it sounds. They also have 99 yen shops with a lot of food. They have green signs with a big 99 on it–hard to miss!
Buses aren’t quite as great (and considerably more difficult for a non-Japanese speaker), but the train system is amazing and fairly cheap (as you know). Definitely do the train thing–it’s an important part of Japanese culture. Depending on how much luggage you have and bother you want to go to, there’s a large price range in transportation from the airport to Yokohama/Tokyo. Here’s my advice: there’s a great site called japan-guide.com that will tell you practically everything you need to know. If you want train schedules there’s quite a few great sites. Here are 2:
Type in Narita on one field and the station closest to wherever you stay in the other field and you should be able to easily find your way around. Seriously: the Japanese train station in the Tokyo metropolitan area is insanely convenient and insanely easy to navigate. It can be crowded but that’s pretty much the only downside. Virtually all the stops will have signs in English; that’s definitely the case for the ride out of Narita. Keep in mind Narita is quite far from Tokyo, and then it’s further on to Yokohama, but it shouldn’t take too long still. There is an express train (the Narita express) that runs between Narita and Yokohama station, but it’s expensive. Really fast though–you pay the premium.
Taxis are very expensive (as a poor student, way out of my price range. I was only in one once, when a rich friend paid.) From the airport, it would be obscene. $100s of dollars.
Not sure about the weather–real JapanDopers can help you there–but in mid-September it was still really hot and very, very humid. Pack light.
About customs–well, there are plenty. But frankly, people don’t really care if you violate them, because you are a foreigner. The only really important one I can think of is: do not walk on a tatami mat floor if you are wearing shoes or slippers; it can damage the floor and is pretty much a no-no. There are others–you’re not supposed to walk while eating, drinking, or smoking, for example. (But people do). And you’re not supposed to place your chopsticks upright in rice… etc. etc. You can read all about that in your guidebook. Just remember, if you see shoes lying around somewhere… you should probably take yours off as well at that point.
Now, if you will be doing sightseeing, you should most definitely go to Tokyo. Yokohama is the second biggest city in Japan, but it can’t compare–nothing can. Also, the big tourist attraction in Yokohama is the Chinatown which would be a little counterintuitive for your first (I’m assuming) trip to Japan. It’s mostly famous for the over-priced food. There’s a ramen museum there though, as well. I never went but it sounded entertaining… I love ramen. There’s also an amusement park with the biggest ferrywheel in the world, if you’re into that kind of thing.
On preview I see Sublight has posted a lot of the same things–but here it is again, if you want confirmation!