It's a little old and on the "light" side, but for a solid introduction to Heian Japanese culture,
Ivan Morris'
The World of the Shining Prince: Court Life in Ancient Japan can't be beat. Here's a
review of it which is a little more descriptive than the Amazon.com stuff tends to get into.
Morris also has an excellent annotated version of
The Pillow Book of Sei Shonagon which is very entertaining in its own right, and a key text for understanding the time period. It's been recently reprinted into a single paperback volume at a fairly affordable price, last I checked.
If you can get your hands on them, the Cambridge History of Ancient China and Japan (not the illustrated single-volume books, which are okay, but kind of lacking, but the multi-volume text-only books) are an extremely good source which both introduce and proceed to go into depth on many pertinent subjects (and point you towards further reading as well) in a very readable fashion.
Those books cost somewhere between 200-300 CAD per volume the last time I checked, so they're not really a buy-it-yourself option for the casual reader, but it's just barely possible your library may have some of them, as mine did.
For technology and scientific discovery in Ancient China,
Joseph Needham's works set the standard. The full length multi-volume set is like the Cambridge History volumes: very expensive and likely out of reach.
But he has a condensed "starter" single-volume introductory work which is well worth picking up. I can't recall the name of it off the top of my head, and I can't find wherever I left my own copy, either. But it should be pretty easy to figure out (probably sort the availbable Needham titles by price and length and pick the shortest and cheapest).
For Japan during the early European contact period to the beginning of the Tokugawa shogunate, Michael Cooper of the Society of Jesuits has a couple of very good books:
Rodrigues the Interpreter: An Early Jesuit in Japan and China, a biography of
the man who was heavily involved in Tokugawa's court (he was the inspiration for the interpreter-priest in James Clavell's
Shogun), and later with Ming officials in China and had a fascinating life.
He actually escaped from a Manchu raid on a Ming fort shortly before the dynasty fell to the newcoming Qing conquerors, and wrote the definitive early Japanese grammar and learning textbook that other people would plagiarize for centuries afterwards.
The other book of Cooper's that I recommend is called
The Southern Barbarians, and it's a little more coffee-table like with lots of illustrations. But still some very good solid information on early European-Japanese relations.
While it's not quite
ancient China, you may be interested in trying a couple of books by
Jonathan Spence set in late Ming to early Qing, which are on the biographical side.
I personally have a copy of
Treason by the Book and
To Change China: Western Advisers in China, 1620-1960, and I keep meaning to read
The Memory Palace of Matteo Ricci, who was one of the major Jesuit advisers to the Ming dynasty court.
Spence writes in a breezily accessible "light" scholarly style, but has extensive footnotes and source references in the back of each book. Although not the sort of "overview" history that you might want to start with, he does incorporate a fair amount on the events and cultural context of the surrounding times.
Hope this helps.