I've not stayed in the Ueno Chisun but From the customer photos in tripadvisor it seems like the same Chisun format I'm familiar with. At 4,000yen a night it is definitely good value.
The empty fridge in the room will help you save money (look around for 2 or 3 litre bottles of your favourite drinks and decant the contents into smaller bottles to take around with you so you don't end up frittering away thousands on the vending machines). Also, by having access to the JR Yamanote line at Ueno you'll end up paying a few hundred less a day on transport than you would by staying at cheaper lodging somewhere like Minowa which gives access only to the Metro lines.
Remember that high end dining is feasible for those on a budget as lunch time prices can be a true bargain. Go for more modest offerings at dinner time, many department stores will offer significant discounts on the deluxe bentos and baked goods just before closing time or there's a bewildering range of curry houses and tonkatsu restaurants and noodle places and all kinds of other delights to choose from at 800yen and under.

I just looked at the Chisun Ueno, and with a 28-day advance booking, it's now ¥5280 for a single. But that's still a very good price. The rooms are tiny, but since you most likely will be using yours only to sleep and bathe, that's not a problem.

From the reviews I've read the Chisun Hotel Ueno seems great. I don't mind the room being small, since I'm not planning to spend much time there anyway. From google maps it looks like a 10 min walk from the Keiseiueno(where the train to the airport goes from, right?) and a little bit less to the subway station which is great. And from the subway map it looks like I can get to the most interesting points in the city with one or two subways, is that right? I'm waiting around a week before I'll book the hotel, since I need to make sure my visa for China is accepted.

For the subway, do you recommend that I buy some sort of day/week etc. ticket or should I just keep buying single tickets? And is the the JR Yamanote line cheaper and/or better then the metro?

As for the food, I think I'll be fine. Wont go to any expensive restaurants anyway. And I'm from Norway, which have a rumor of being the most expensive country in the world, so I guess Tokyo will be around the same. Anything else then sushi that is a must eat in Tokyo?

Keep coming with suggestions, I really appreciate it!

Oh, I was in Norway last summer, and Tokyo is definitely cheaper! If you can't find an evening meal for less than ¥1500 or about 100kr, you aren't really trying. And lunch and breakfast are cheaper.

My usual recommendation for transportation is to buy a Suica card, which is a prepaid card that you can use on all trains, subways, and buses that you are likely to ride. Sometimes the train is more convenient, and sometimes the subway is more convenient. It depends on your destination.

The Suica cards are sold in units of ¥1000 with a ¥500 refundable deposit, and if you run low, you can always add more money to the card at any station.

As you enter the platform area, tap your Suica card on the designated spot on the turnstile, and tap it again when you leave. (In other words, it works like London's Oyster Card.) It automatically figures the fare.

If you can find the book Eating Cheap in Japan, it has very good descriptions of traditional foods.

I like okonomiyaki (somewhere between a crepe and an omelet, filled with meat or seafood and grilled in front of you), tonkatsu (breaded pork cutlet, but much lighter than its European counterpart), soba (buckwheat noodles), yakitori (chicken shish kebabs), kamameshi (like a seafood risotto), tempura (breaded and fried seafood or vegetables, again, lighter than European breading) and curry rice (different from Indian curry). But it's fun just to walk around and see the plastic models displayed in the restaurant windows.

For good sushi, either go to the fish market area or to the fish section in the basement of a department store. There is usually a little sushi bar hidden behind a curtain.

Keisei Ueno is the terminus for Keisei main line. That is where you will alight if taking Keisei Sky Liner, City Liner or limited express. It is possible to take the underground passage all the way to the east side of Chuo Dori (exit 1 of the Tokyo Metro Hibiya Line Ueno Station), but the elevated walkway is more scenic.

Agree with buying an IC card like Suica or Pasmo. The biggest benefit IMO is it saves you the time and trouble of looking up fares and buying single trip tickets, which may be overwhelming for people not familiar with the concept.

Generally speaking, JR fare is slightly less expensive than the subway.

If you are looking for "good value" sushi then I recommend the Sushi Zanmai chain:

http://www.kiyomura.co.jp/shop/shop21_01-00.html

Ok, sounds like I should buy a Suica card, do they work on the Keisei Main Line as well? And is it different prices on the subway depending on how far you are going? What does a normal trip with the subway usually cost?

Yes X 3
Plug your journeys into http://www.hyperdia.com/ to find the fares. Sometimes a route that uses a mix of railway companies turns out to be quicker than a cheaper route that sticks to one company's lines.

Yes, Suica cards can be used at Keisei stations, but for convenience's sake time, you should just go to the Keisei station's counter or vending machine and buy a Pasmo card there. Suica, issued by JR East, and Pasmo, issued by the consortium made up of subway and private rail companies, may be used on each other's networks and even use each other's ticket machines to add value. Generically they are called IC cards and many stores now accept them as form of payment.

Train fare is determined by distance. A subway (Tokyo Metro, Toei, etc.) trip generally costs between 160 yen and 300 yen. JR East trip costs from 130 yen and up, as they have a more extensive rail network with limited express and shinkansen options.

You can also try conveyor belt sushi.

Suica/Pasmo cards are handy precisely because the fares vary by distance.

In the old days, you used to have to look at the map on the wall, find your starting point and destination, figure out the fare, and stand in line at a machine to buy a single ticket.

Uniformed men stood at the turnstile, punching holes in the tickets. The constant clicking almost sounded like castanets.

Next came single tickets that you inserted in the turnstile to have them punched automatically.

Nowadays, you just tap your card on the turnstile and it automatically figures the fare.

I prefer the subway system. As a means of transport, sushi conveyor belts tend to be slow, and limited in coverage area.

But as long as you only ride the light green plates it's pretty cheap.

Okey, thanks for the answers!

I don't usually plan exactly when to do things, but just to have an idea about how much time I will spend on each place I will set up a suggestion here, that you can criticize:

Monday:
-Arriving at Narita, keisei main line to ueno. Then walk to the hotel and just check out the area around the hotel, grab some dinner.

Tuesday:
- Walk to Asakusa in the morning, see the temple. Maybe walk down to see Edo-Tokyo Museum
- Take the subway to Chiyoda, check out the imperial palace and the area around it.
- Walk to Akihabara and stay there for the rest of the day

Wednesday:
- Take the subway to Odaiba (if possible) and see the buildings here
- Take the subway to Shinjuku, check out the view from tokyo goverment building and explore this area

Thursday:
- Walk to the Yanaka part and see the market
- Take the subway to Shibuya. See the shrines at Harajuku and hopefully some youth culture
- Maybe walk to Minato, see the capsule tower, pokemone center, Sengakuji etc.

Friday:
-Eat breakfeast, then go back to the airport with the keisei main line

So I think this will give me good time to see each part of the city. But is it to much or to little? Any important places I've missed? I also want to see Ghibli Museum, but that might be a bit out of the way?

That is a pretty long walk from Asakusa to Ryogoku. I'd recommend the subway. You will have to change lines at Kuramae.

2.4km according to google maps. And that is no problem for me, so I would prefer that to having to change subways etc. Also the walk is next to Sumida river so hopefully it will be a nice view!