Hi. I will just answer based on my own experience as best I can. I currently live in Japan and have travelled throughout Japan.
1. If you plan to stay in cheap places, it’s probably going to be about an average of ~$25-35+ (¥2500-¥3500+) per night at hostels or capsule hostels. I think it might be the capsule hostels that you are thinking about, with the pods. There are many in Tokyo and Osaka, as far as I know. I don’t think that there are any in Kyoto, but maybe a couple if you search. For food, I feel that it really would be a pity for you, if you don't at least try out the kinds of food in Japan. Japan is known for its unique food. But if you average about $10 per meal, then that’s about $30 per day. This is if you really are frugal about it. If you go to gyuudon places (beef bowl places), meals are typically only $3-$5 with a big bowl of rice and beef or chicken.
Kyoto would be your best bet for temples and shrines, and souvenirs are everywhere. Entrance ranges from ¥500-¥1000. Plus, there’s transportation too.
So if you put it to be ¥10,000 (~$100) per day including all costs:
1 week = $700 or ¥70,000
2 weeks = $1400 or ¥140,000
And so forth. (this is being on the cheap, safe side)
2. Cities like Tokyo and Kyoto are very hard to get around if you are walking, because they are very big, especially Tokyo. All of these cities typically have day passes for trains, subways, and/or buses. And not all cities have bike rentals.
Go to: http://www.japan-guide.com/e/e623.html
They have summed up lots of things in Japan and have listed many ways of how to travel around major cities in Japan. So you’ll be able to do some thorough research of the cities you want to travel in.
3. If you want a hired Japanese tour guide, this is going to be expensive. The best thing to do is to find a person who will do it for free. Again, japan-guide.com is a good place to look.
http://www.japan-guide.com/local/
Look at the “Services” section and click “Travel Guide”. They will sometimes do it for free if they have the opportunity to gain a foreign friend. OR, try to become friends with a Japanese person BEFORE you come to Japan. Then they will be able to show you around. If you are lucky, they may offer you accommodation for free, and that will reduce your travel expenses.
4. Japanese people are used to foreigners, but they will stare at you if you are not Asian and “blend” in. Just some things to take note of, not typically for reputation, but just to know in Japan:
a) Don’t speak loudly in trains or subways, or buses. (although even Japanese people do sometimes)
b) When in Japan, coins become numerous, so make sure to have a coin pouch.
c) Restrooms may not have soup or paper towels.
d) places usually don't open till about 10:00am in most cities. (9:00 is usually the earliest)
e) don't try to stand out.
Overall, when in doubt, just look at what Japanese people do, and do what they do.
Other ppl prob have better tips. I'm so used to japan that I can't think of anything now.
5. Storage will be pricey. If you stay at hostels, even if you can’t check in yet or have checked out, they will typically let you leave your stuff there for free or keep them in a locker for a fee. This is the best way. In all stations in Japan, there are lockers where you can keep things for the day. A quote from Japan-guide:
“The cost of coin lockers is based on calendar days (midnight to midnight) and is typically 300 yen for small lockers, 400 yen for medium lockers and 500 yen for large lockers per calendar day. In other words, if you use a small locker overnight, you will have to pay an additional 300 yen when picking up your luggage the following day. Lockers are emptied by station staff after three days.”
http://www.japan-guide.com/e/e2274.html
I recommend you just store your things at whatever place you are staying. The only other option is to have a big backpack. (But it will be a pain) Also, don’t plan to ship anything home, because shipping costs are ridiculously expensive. Maybe a small package less than 2kg might be around ¥1000 or $10.
6. They aren’t bad in Kyoto, but Osaka and Tokyo are very crowded. As long as you know where you are going, you’ll be fine. Just don’t stick anything where the doors close, and try to stay as close to the doors as possible, especially if you need to get out. You can sometimes be stuck in the back and miss a stop if there are a lot of people.
7. It’s safe in Japan. Any country will have crime. I’d say Tokyo and Osaka are probably the cities with more crime, and sometimes dangerous in areas where there are foreigners, b/c people do target foreigners. If you are a girl, it’s more dangerous. But it seems you aren’t, so that’s better. The best thing to do would probably travel in a group and/or not at night if you are alone. Stay away from shady places too. Try to travel lightly (no giant backpack if possible), and don't give that "tourist" vibe, like looking at a travel guide book, looking lost, or looking like you don't know where you are going. In Tokyo, people might target you. And you don’t have to worry about pickpockets as in Europe. (where it's crazy)
8. Getting sick while overseas is just something that is normal because of a different environment, jet lag, or different food. Japan has many hospitals and clinics, so if you do get sick, you’ll be able to find a place to get treatment. Just make sure you have health insurance back in your own country to get reimbursed later. Being up to date on flu shots wouldn’t hurt.
9. This is the thing. In order to meet other people around this age, alcohol might be a necessity. The best thing would be to travel with other people this age (you can usually find people at the hostels you are staying at, who are also travelling alone). But clubs or drinking areas like at rivers are good places. Bars and restaurants, shopping districts, and places like this are good. Since you plan to travel young, and the drinking age is 20 in Japan, if you are not AT LEAST 20 years old, then don’t plan on meeting up with many women as they will not be interested if you are not at least legal. If you are legal age when in Japan, Japanese women are happy to hook up with foreigners. 18 or 19 year olds are usually still in high school or first year in college, so wouldn't be out anyway.
10. Well, Japanese is a syllabic language, with not much tone like in Chinese or other languages. This means pronunciation is not very hard. Learning basic phrases is the best way, and learning the first 2 writing systems (hiragana and katakana) will help you tremendously. Make sure to know basic greeting, how to order food, ask how much something is, numbers, how to say “help”, etc. Most Japanese can usually understand a little bit of English if you speak slow, so you should be fine. Plus, there is more English usage in Japan than one might think as they use a lot of English words with no Japanese equivalent, with their katakana words.
11. If you travel to Tokyo, expect snow. Kyoto or central Japan usually doesn't have snow. But it will be cold. Most things will be open, but you just might not be able to see any big events like festivals which typically occur on the summer. And while you say you aren't interested in the cherry blossoms and such, temples and shrines are at their best when the trees, flowers, and plants are in full bloom and at their best. But you'll still see and appreciate the places even without these things with the bare trees and no flowers in winter.
12. Hmm. I'd say it's safe. I've been in japan for over a year, and I'm still alive with no health problems. But just to be safe, don't travel in the Fukushima or Sendai area, and you'll be fine. Since you'll only be in Japan temporarily, there's nothing to worry about. Some ppl are overly paranoid.
13. No ninjas or samurais. You might find some ppl cosplaying, but that's it. There are still some geisha around, but not many. Some are in Kyoto around the Gion district.
14. Well, you need to be at least 18 to travel by yourself. Travel as lightly as you can, b/c having a big luggage will always be a hassle. Plan where you want to go and what cities you want to visit with a rough itinerary. Save up as much money as possible. Try to get at least 1 or 2 credit cards before you travel to Japan, for emergencies. (I first came to Japan without a credit card, and it was not a good idea, especially when I ran out of money). Try to make some japanese friends to show you around once in japan.
And traveling alone is always boring. It would be good if you can have a friend, sibling, or relative to travel with you. Even if traveling by yourself would be a worthwhile experience, traveling with someone, especially in a foreign country is always really nice.
Good luck.