Любая электронная карта вам в этом поможет, сейчас они снабжены маршрутами общественного транспорта.

As far as I know, the Bridge is not open for walking during normal times. It is open once a year during the Istanbul Marathon during every fall. That time they close it to the public and open for running & walking.

На велосипеде через мост бесплатно?

Try the public tram system.I'll try to find out which station you need to alight. Also depends where you're travelling from but the trams are really good.

The Bridge joins the european and asian side of İstanbul/ Turkey. Both sides are in the same city