Whoops, I had not thought that you'd have the Limo tickets! The following excludes use of Suica prepaid card then. From Shinjuku to Asakusa, it takes a bit longer but there is a Metro-only route to get there. (Take Metro Marunouchi line to Ginza, then change to Ginza line to Asakusa.) You can get back to Shinjuku the same way, Metro only. Shorter way will involve JR line.
OK, to go to Kamakura, sorry, you have the less expensive choice compared with JR:
http://www.japan-guide.com/e/e3115.html
So you can buy this (non-JR, non-Metro) pass to go to Kamakura from Shinjuku. (JR train costs more, and is not covered by Metro pass anyway.)
When you come back from Kamakura using the above pass, you will reach Shinjuku - from there, if you want to go Metro only, you can go Shinjuku - Ginza (Marunouchi line), then Ginza - Akiharabara (Hibiya line).
If you are planning all this in one day, it sounds a bit tight, though. Actually, Asakusa and Akihabara are relatively close, so it might make sense to allocate one day to this area (if you want to do some shopping there), and save Kamakura trip for another day.
The above is completely without the Suica. If you decide to go Nex & Suica + JR Pass, then:
- You can use JR + Metro to go from Shinjuku - Asakusa (faster than Metro only), then back.
- You can buy the Kamakura pass (please look at the link above) for the Kamakura portion.
- From Shinjuku, you can take JR to get to Akihabara.
And yes, if your JR Pass is valid for the day when you travel from Tokyo to Narita, you can use that for Narita Express as well.