Let's say I brought my bicycle to Japan and want to reach central Tokyo quickly and easily. Can I just ride Tokyo Monorail with my bicycle along?
Yes, with no additional fee, but you must fold your bicycle and put it into a bicycle bag.
You can not enter the train with a bicycle as-is.
If you want to bring it, you must disassemble the bicycle (front wheel, or fold it if it is a foldable bicycle) and put it into a bicycle bag made for that purpose, then enter the train.
Article about this bag, in Japanese: https://ja.wikipedia.org/wiki/輪行袋
If you had your bicycle when landing at Haneda, chances are you have such a bag already. Otherwise, you can buy one for between 1300 and 10,000 yen.