Would the JR Sobu Line Rapid train to
Shimbashi be a good option? Also, on the following route, there are no change of trains involved, right?:
http://xs113.xs.to/xs113/07126/NRT-Shimbashi.gif
Is Conrad within a walkable distance from the
Shimbashi station? If not, how much would a taxi ride cost?
How would the options compare to taking the Sobu Line to the
Tokyo station + taxi to Conrad (which should be approx. 1200 JPY, right?)
Any advice would be appreciated
Yes, the Sobu train is direct, and you can walk to the Conrad from Shimbashi. It's about 5-10 minutes. The thing is that it's a bit hard to navigate in that area, and I believe you have to cross some foot bridges. The Sobu line platforms at Shimbashi are also basically in the seventh level of Hell, and you'll have to deal with steps etc. just to get out of the station. So if you go that route, have your heavy luggage delivered, and only take a small overnight bag with you.
If you're going to do that, it may be faster and just as cheap to take the Keisei Skyliner or limited express (whichever is departing first) to Nippori, then change to the southbound Yamanote Line, which also goes to Shimbashi.
If you'd rather keep your bags by your side, then the limousine bus is the best option, followed by taking NEX to Tokyo Station and a cab from there.