^I went to Osaka/Kyoto June of last year and the weather was mild, although there was a bit of rain showers here and there. I did a fullday tour of Kyoto from Osaka and I must say, the tour covered a lot of the must see places like the golden pavillion, osaka castle, etc. but I'm sure there's so much more to see and do! Another full day would've been perfect!
Not sure about accomodation but pretty sure there are hostels in both cities. As for food, price range can be cheap to ridiculously expensive. I think if you're on a budget, you can't go wrong with ramen and sushi-go-round places and konbinis. But do yourself a favor and splurge a bit on food coz it's gonna be worth it. Figure around at least 200 USD for good kobe beef (set meal at Rio Teppanyaki), $20 per 5 pcs otoro sashimi and about the same price for amaebi sushi (I'm drooling while typing...)
On the lower end, I think you can get sushi set for less than $10 at a konbini and $1 sushi plates at the conveyor belt places. Haven't tried em but hey it's Japan, bet the quality is still good, if not better than what we got here in Chicago. In Osaka, cheap finds are takoyaki, okonomiyaki, and kushiage of all sorts. Nothing more than 10 bucks. Japanese beer is a must try also, kirin, asahi, suntory, etc. maybe 4-7 bucks a glass
Transpo-wise, always get a well situated accomodation to save on cab/subway fare. What I love about Japan's subway system is that it's relatively cheap, extensive, and always punctual. I got the suica card and loaded it with say $15 to get around Osaka for 2 days. There's really not much to see there but if you're a foodie like myself, it's heaven! I payed like $50 for r/t airport limo bus that dropped me right in front of my hotel. The Osaka-Kyoto tour was like a little more than $100 but lunch was included. getting around in Kyoto might be more challenging though...
As for shopping, I found that Osaka prices are somewhat cheaper than Tokyo. Found a lot of shoes under $20 at Shinsaibashi suji and clothes under $10, which is hard to find in Tokyo even if it's Jan. (sale season). As for souvenirs, cheap must buys for me: make ups from matsukiyo, japanese magazines, regional goods like takoyaki flavored snacks (glico),etc. Nothing is more than $10.
But if money is no object, Japan is a haven for shopaholics too! Just skip the foreign brands as they are always more expensive there than anywhere else in the world. So I often buy brands exclusive to Japan like Burberry blue label, Garcia Marquez, Cecil mcbee, etc. and beauty prods. like chacott, palty, etc.
Hope this helps! Love Japan to bits! I'm sure you will too!