Well i stayed at
Economy Backpackers Hotel NEW KOYO wich is 2700 for a single room (really small though) and 4800 for double room. It is a 10 minutes walk to the Station and from there 2 stations to Ueno.You can actually walk to Asakusa in 25min. To the next JR Station ist also around 10 min (maybe less).
The Hotel itself ist quite ok, clean, the staff is very helpful, internet runs at 600yen an hour or 300 yen if you bring your laptop. i was lucky as i could use the w-lan of some neigbouhr in my room.
I think that the roomsize is only made for a short stay and that the double room is actually very small for 2 people, but for everyone on a budget its ok. They have showers wich u can use 24h and a small kitchen and there is a Japanese stile bath wich you can only use to certain hours (2-3h a day not mixed). There are only 6 double room i think, so you should book well in advance. At the moment i would even recommend you to book in advance for a single room.
There is a convienince store nearby and a supermarket.
you can get there from Narita Ariport for only 1160yen if you are alright to walk around 3 min to change from train to Subway and around 10 min from the train to the hotel. You will take around 2 hours to get to the Hotel. I you choose the faster train it will cost you 900yen more and will save you between 20 and 30 minutes depending on how the trains are running.
Good luck