We are going to Japan on March 2017 (Osaka, Nara, Kyoto, Kobe, Hiroshima, Miyajima and Tokyo). We are going to spend about 2.500 Yens per night . The price is usually high for the weekend, but we found a Hostel at Kyoto that doesn't overprice during the weekend.
In Osaka there is a 3.000 or 2.300 yens pass that allows you to use all the public transport of the city (not JR), and visit MANY attractions for free (including some short cruises), plus some other attractions with a small discount (for us these discounts doesn't worth it). The pass name is Osaka Amazing Pass. If you choose the 2-day pass (3.000 yens), you can't use all the transport of the city (but more than enough, maybe sometimes you will need 10 more minutes of subway, but not more), while the 1-day pass (2.300) has some more options for the transport (for instance, it includes the transport from the airport to Osaka (but please, double check this if you are going to but the 1-day pass, I could be wrong!!!!), which costs 920 Yens).
After comparing prices, we decided to get the 2-day pass and to pay the 920 Yens from the airport to Osaka. That would be 3920 Yens, for 2 days, with free entry to the best attraction of the city (except for the aquarium, which is a bit expensive). We are staying at a hostel (2.500 Yens per person, named Hostel Zoo), but there is an option for 1.100 Yens per person (hotel Diamond). Both of them are located near each other and near the train/subway.
So, in our case, is 920 (airport-city train) + 2.500 x 2 nights + 3.000 2-day city pass = almost 9.000 Yens (4.500 per day) staying at a decent hostel (to be honest we are staying 4 nights because we are going to the Universal Studio Japan, so the second day in Osaka we don't have to take a train, that will leave us more time to visit more attractions. The 4th day we are going to wake up early to visit Nara, and we will sleep at Kyoto). Also you have to add the food, but if the 1.100 yens per night hotel is ok for you, that would be a bit less than 3.100 Yens per day + food. Also, you can decide to but only the 1-day pass and stay only 1 day here (1.100+2.300+food), but I think that you will enjoy more this city for 2 days.
In Tokyo (near Asakusa Station), like 3-4 years ago we found a 24H supermarket called Rox, where you could find a lot of lunch boxes for less than 400 Yens, and at night, many of them with a discount (I bought a HUGE box of sushi for a ridiculous amount of Yens... I remember that I needed to have a 2 hours break to finish the food. I can't remember the price, but it was for sure a bargain). I don't know any place like this in Osaka since I have never been there, but I doubt that you can't find something similar if you ask at the hotel or google it for a while.
I can't provide feedback for the hostel/hotel that I mencioned, but at some website the Hostel Zoo looks fine, while the Diamond looks a bit dirty for my standards, but if the money is a problem, I think Diamond is the best option (compare opinions and try to find other options, there could be better places to sleep in Osaka).
I hope my English was ok, feel free to ask me anything ^^
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