When is the best time to go?

Spring (last week of March/1st week of April) for cherry blossom season or fall (last 2 weeks of November). worst time to go is in the summer, rainy season and humid, close to 100% almost everyday.

When is the cheapest time to go?
depends, but I sometimes see find cheap fares beginning of march and beginning of october

Does NRT vs HND make much of a difference?

yes. nrt is very far from tokyo but flight times are usually much more convenient. hnd is much closer to tokyo but usually arriving very late and leaving very early. i'm flying into hnd for my next trip bc i get get a business class delta ticket for 120,000 miles while for the same dates to narita it was 120,000 for economy. i have to leave at 655am, but it's worth it to me.

What can I expect to pay in airfare?
800-2000

What neighborhoods are the best to stay in?

depends what you wanna do see/do and your budget. shibuya, shinjuku, ginza, are good places to start.

What are my transportation options getting around Tokyo?

incredible train and subway system should get you just about anywhere you wanna go. also buses, taxis and even some boat rides.