Cycling is generally very safe, particularly in Taito-ku. There aren't any bike lanes really but you're free to cycle on the pavement, as long as you dismount when it's too crowded. Furthermore drivers are used to cyclists, just keep tight to the left on bigger roads. Helmets are seldom worn, even by mothers with a kid on the front, one on the back and one strapped to their chest.
Parking is a bit different. If you lock your bike up (not that it's really necessary, generally it's very unlikely your bike will get stolen) just anywhere, you'll be either asked to move along or get a warning ticket stuck to it. Leave it too long and the council will lift it. Even designated parking areas (usually outside stations) often require a residency tag / pass from the council. As such I usually park outside convenience stores - allowed, and no chance of it getting moved.
Police do not look kindly on headphone or mobile phone usage whilst cycling either - you'll likely be stopped (though they're very polite and will do their best to explain in English).
Other than that, Tokyo is great to bike around! Perfect way to take in the sights, particularly within the Yamanote loop.
(Fellow Edinburger in Tokyo :))