Japan for the first time: only 13 days

Hi!
I planned my first trip to Japan in September!
I'm going with a friend of mine. Now, because we only have 13 days I would like your advice to what we absolutely need to see, and what wouldn't be worth it :)

The daytrips can be changed because I'm not sure when it's best to visit what ^^;
We are first going to Kyoto as a friend of mine lives there and will guide us 2 days probably. (and so we will have someone to help us around the first 2 days :) )

Our schedule so far:
Arrive: tuesday 9am at Narita airport. Take train to Kyoto.
Wednesday: Kyoto
Thursday: Kyoto
Friday: Osaka (probably, can switch)
Saturday: Nara (probably, can switch)
Sunday: take train to Tokyo
Monday: Tokyo
Tuesday: Toyko (Ghibli museum)
Wednesday: Mount Takao
Thursday: Kurukama
Friday: Tokyo
Saturday: Yokohama
Sunday: Tokyo, perhaps we're going to Narita and stay there or leave from Tokyo in the morning on Monday to catch our flight...
Monday: Flight back at Narita at 10:30.

As for interests, my friend is a huge ghiblifan and likes birds and nature (biologist ;) ) I love traditional architecture (studying monument and landscape conservation),nature, Arashi and shopping.

Do you have any advice for us?

I forgot to say but we're planning to buy a 14day JRpass.
I also booked a hotel(for 3 days) and a ryoukan (for 2 days) in Kyoto so that will be our main place to stay. I still have to book for Tokyo. Probably going to get 2 beds in a female dormitory to safe money :)

You have 13 nights to stay in Japan, and will take Japan rail pass 14 days. I think you might better become more ambitious to take long distance rides on JR trains, especially bullet trains. This is just a starting itinerary.

Kyoto 3 nights
Hiroshima 2 nights
Matsumoto 2 nights
Sendai 2 nights
Tokyo 3 nights
Narita 1 night

Mt Takao might be interesting, but you might better visit higher mountainous area such as Kamikochi (near Matsumoto). Osaka and Yokohama are like Tokyo and Kamakura is like Kyoto. Nara is somewhat special but it is somewhat similar to Kyoto. Since this is your first trip to Japan, you might better maximize the value of Japan rail pass.

September is, honestly, not the best season for traveling Japan. It is still hot humid summer of Japan, and you might encounter typhoon or heavy rains. Therefore I recommend you to base in rather big cities for more than two nights each, they you can have flexibility in your itinerary. For example, if the weather is nice in Matsumoto, you can take a day trip to Kamikochi while if it is rainy you can stroll Matsumoto town, visit Matsumoto castle and Asama onsen.

Have a nice trip!

Your itinerary as it is looks fine, not rushed.
If you stick to your current itinerary, you only need 7-day JR Pas to cover Narita - Tokyo - Kyoto - Osaka/Nara - Tokyo. In and around Tokyo, you don't need the rail pass; you can go with prepaid card (JR Pass makes sense only if you make long-distance, inter-city rail travel).

You "could" put in a day trip from Tokyo to Nikko (using Tobu railway), or to Hakone (using Odakyu railway). Both locations have abundance of nature around, if you are interested.

For Thursday back in Tokyo, I assume you mean "Kamakura"? :)

yes I meant kamakura ^^;

Thankyou frog1954, I'll have a look at those cities and see what we can visit. Our plan was perhaps to visit hiroshima too so that would be nice if we could include that as well!
Thanks, guest! I didn't know we don't need the JR pass around Tokyo! That saved us some money!!

For now, I have two options for schedules:

BETTER SCHEDULE IF WEATHER IS BAD + MAKE MAX USE OF 14 day JR RAIL PASS
Arrive: tuesday 9am at Narita airport. Take train to Kyoto.
Wednesday: Kyoto
Thursday: Kyoto
Friday: train to hiroshima
Saturday: hiroshima
Sunday: train to matsumoto (kamikochi = higher area)
Monday: matsumoto
Tuesday: train to sendai
Wednesday: sendai
Thursday: train to tokyo
Friday: Tokyo
Saturday: tokyo
Sunday: train to narita in the evening
Monday: Flight back at Narita at 10:30.


SSCHEDULE WITH 7 DAY RAIL PASS (NO NEED FOR JRPASS IN TOKYO)
Arrive: tuesday 9am at Narita airport. Take train to Kyoto.
Wednesday: Kyoto
Thursday: Kyoto
Friday: Osaka (probably, can switch)
Saturday: Nara (probably, can switch)
Sunday: take train to Tokyo
Monday: Tokyo
Tuesday: Toyko (Ghibli museum)
Wednesday: Mount Takao (can change)
Thursday: Kamakura (can change)
Friday: Tokyo
Saturday: Tokyo
Sunday: Tokyo, train to narita in the evening
Monday: Flight back at Narita at 10:30.

(why is there no edit button!)
If forgot to say, perhaps it's not that good to travel from Matsumoto to Sendai (seems like it takes a long time) and then back to Tokyo... Anything else interesting around there?
perhaps mount fuji?

perhaps it's not that good to travel from Matsumoto to Sendai (seems like it takes a long time) and then back to Tokyo... Anything else interesting around there?

It is, in part, because I am a train freak :)

It is September and the weather in general is not so nice. Therefore you might better go north to avoid hot weather. Sendai is the center of Tohoku area and you can day trip to
1. Morioka and Hachimantai
2. Chusonji in Hiraizumi
3. Matsushima
4. Yamadera.

http://www.japan-guide.com/list/e1102.html

As for Mt Fuji, you may be able to see it if you take the Chuo line (via Kofu) limited express train. If you go first to Nagano and take the Nagano Shinkansen, you will be able to see Mt Asama which is also nice.

What I posted is just a first draft, and you can choose from many alternatives. I just tried to maximize your rail pass. However, you might better consider the hot weather in September.

In any case, planning the itinerary is by itself a pleasure of travel. Enjoy planning!

Okay so I changed our schedule!
We'll go for a 7 day railpass

Arrive: tuesday 9am at Narita airport. Take train to Kyoto.
Wednesday: Kyoto
Thursday: Kyoto
Friday: train to Hiroshima
Saturday: Hiroshima
Sunday: train to Osaka
Monday: Osaka, train to Tokyo
Tuesday: Toyko (Ghibli museum)
Wednesday: Mount Takao (can change)
Thursday: Kamakura (can change)
Friday: Tokyo
Saturday: Tokyo
Sunday: Tokyo
Monday: train to narita in the morning because our flight back is at Narita at 10:30.

But I'm wondering if it's smarter to stay 4 days in Kyoto and then do a daytrip from Kyoto to Osaka. Then leave Kyoto for Hiroshima, stay there 2 days and leave Hiroshima for Tokyo. Instead of doing Kyoto-Hiroshima-Osaka-Tokyo, because that means I'll need to book a hotel in Osaka for just 1 night. Any thoughts?