If you are just staying in Tokyo and the Tokyo area the JR pass is a humongous waste of money.
But if you are going to another city like Osaka and back to Tokyo then the JR pass makes sense economically.
Harajuku, Shibuya, Ueno, Akihabara, Ginza (via Yurakucho Station) you can go by JR train.
Roppongi - NO JR train, but very easy from Shinjuku if you take the Oedo Subway Line
Tsukiji fish market - No
Asakusa - You can take JR train to Kanda or Ueno Station and change to the Ginza Subway Line there to Asakussa.
"Rainbow Bridge" (I think you mean Odaiba), you can take JR to Shimbashi Station, where you can change to the Yurikamome Line which crosses the rainbow bridge.
To answer your question, is it costly?
No, single fare tickets in Tokyo cost roughly 130-250 yen one way per ride depending on distance. Somtimes private rail and subway end up cheaper than JR fares, just depends on the route you take.
The only costly route is crossing the rainbow bridge to Odaiba, those fares tend to be a little higher than average because of the cost to build those rail lines. One way from Shimbashi to Daiba Station across the rainbow bridge is 310yen.
If you know the exact name of stations, then you can calculate routes and fares using hyperdia.com note you must know the start and end station names exactly.
Edited: 5:33 pm, September 20, 2012