April should be the cherry blossom season so it'll look nice, but accommodation will be more expensive and possibly harder to find.
1. A Bus will be cheapest but the train would be the most convenient.
http://www.kansai-airport.or.jp/en/access/index.html
Bus is 4000 for a return ticket.
Train is about 3300 each way on the Haruka express. (depends on seat, unreserved is cheapest.)
2. The big places to see are Kiyomizudera and environs (My favourite shrine is there, Jishu Jinjya). Kinkakuji. Golden Temple. (Don't make the mistake of going to Ginkakuji, which is nice and all but totally different.). Ryoanji the famous rock garden. Gion. where the geisha and maiko are. The back lanes are better for a glimpse of a geisha rather than waiting outside a restaurant with other tourists.
Heian shrine and gardens. Nijo with it's ninja-proof singing floors.
I've never been (inside) but it's relatively easy for foreign tourists to get on a tour of the Imperial palace so I'm told.
Kyoto Train Station. There's a good view of the city from the roof, and it's an impressive piece of modern architecture.
More out of the way is Taisha Fushimi Inari a mountain full of torii tunnel like paths.
I'd recommend a day side trip to Nara to see the Dai-Butsu.
I'd also recommend Japanese bathing. Either in the Big Bath at a hotel or visting an onsen. Kuruma is 30 mins on the train from Kyoto has a nice mountain temple and several onsen with rotenburo, a nice way to relax after sightseeing before heading back to Kyoto.
Get up early to sightsee before the busloads of tourists arrive.
3. hard to say. The Yen is very strong at the moment (or Dollar, Pound and Euro are very weak. 1000 yen is about $13 today)
I'd say you'd need about 4000 to 5000 yen a day for food, entrance fees, and bus pass. (the bus is the best way to travel in Kyoto, a daily pass is 500 yen. The subway isn't very big and doesn't go to tourist spots, so isn't worth the extra 700 yen really.
http://www.city.kyoto.jp/koho/eng/access/transport.html)
Accommodation would start at about 2500 a night for a dormitory (K's House, Tomato Guest House) Business hotels single room start about 5500 a night. (look at
http://travel.rakuten.com/ for ideas) I'd recommend finding somewhere near to Kyoto station for convenience when arriving and leaving.
Have a good trip.
i have just a few questions.
Agreed they're rather different. I've been inside gosho (Imperial Palace) and you do need to book but I enjoyed it.