Hi Wex,
I would indeed recommend using your bike to travel to TSUDANUMA and use a monthly pass from there (which is ¥ 13,300). This is by far the more economic option, because Yakuendai is served by the Keisei Railway which is a different company than JR and transferring all the time can cost a lot of extra money. By bicycling you can save about 5,000yen on a monthly basis.
Alternatively you could travel by bike to TSUDANUMA and buy a JR Tokunai pass from there (¥ 730 yen) these are day passes which allow for unlimited travel in all 23 wards of Tokyo on JR lines. This would cost more than a money pass Tsudanuma - Akihabara but is still cheaper that buying single tickets for a return each day (¥900). The great advantage here is that you can see all of Tokyo at the same time.
Here's a great video by Gajinpot explaining the Tokunai ticket.
Hope this helps!
Daniel