When's the best time to go?
-- there is a big swing in weather - around June-July it gets rainy, then after that it gets hot and humid as hell.
Its hard to plan - my parents came in december last time and it was pretty much perfect, but depends on your weather tolerance i guess.
What cities should I check out?
-- Kyoto, Osaka, Tokyo are pretty much the staples.
I'd suggest going up to the mountains - again, season depending, places like Niigata are excellent off the chart getaways in Summer (the ski slopes become water parks) , and in winter, the ski scene is great.
How long should I stay?
-- if you can stretch to 2 weeks , that's ideal, but that's a burden. It's best to enjoy japan at a slow pace.
Best places to eat at?
-- doesn't really matter, the food quality here is off the chart.
if you want cheap and cheerful theres a ton of places, my personal fave being chain Kohmen's insanely awesome Kogashi tantan men.... add in the dried garlic extra and it's mouthgasms
other options are available, but in terms of food ? you're good. unless you are an epicure looking for the top end , you won't leave disappointed.