My experience is from a few years ago (five) so some things may have changed.
Yes, I was advised to take small packets of Kleenex, a small bottle of hand sanitizer and a handkerchief. I found many beautiful handkerchiefs for sale the moment I landed in Kyoto, used one throughout the journey and brought back some unused ones for friends at home -- beautiful colorful maple leaves with an occasional outline in gold, a cat pattern, a rabbit pattern -- and all were gratefully received.
I made use of the Kleenex and the hand sanitizer and was grateful for the suggestion.
By the way, if you find yourself seeking a Western toilet, look for the handicapped sign. That was a great boon for my aged mother.
As for gifts, I brought local products: pounds, half-pounds and sampler packets (four kinds on one box) of Hawaii coffee; a couple of boxes of fine Hawaii chocolate; and about two dozen small sampler packages of macadamia nuts.
The latter went to a person who did a small favor, a small child who walked by (with his mother's permission first), a policeman who went out of his way with directions, etc. The larger items were for hosts in several cities. At the end of the trip, I remember wrapping up my small pastel fleece blanket (very useful on air conditioned buses and trains) to our tour guide for his granddaughter (in addition to the tip collected from our entire group).
I utilized computers at hotels to check and send e-mail. Be sure to have your passwords with you and clear the machine before you step away. If I were to go today, I would take my iPad if only for the photo and note taking capability. I am not certain, but I believe I recently heard something about Verizon Wireless having a plan for international use that didn't cost a beloved limb.
Have a wonderful time and post photos when you come home!