What would be the best way to do this?Take the JR Tokaido Shinkansen from Kyoto to Shin-Yokohama Station. All trains between Kyoto and Tokyo stop at Shin-Yokohama, from where you can take a train or bus to your final destination in Yokohama.
We have our JR passes for the main trips, but we're unsure whether these will work for short trips in the Yokohama area and to and from Tokyo for day trips.
There are some JR train lines that are helpful for getting around Yokohama, especially the JR Negishi Line. But on all non-JR transportation, the Japan Rail Pass will not be valid. About transportation in Yokohama:
http://www.japan-guide.com/e/e2360.html
We will be staying at the Mystays hotel in Yokohama, which does have an underground station near by which connects to Yokohama Station, will JR passes work on this line?
No. The hotel is located close to the above-ground Koganecho Station on the Keikyu Railway Line and underground Bandobashi Station along the municipal subway line. Both lines get you to Yokohama Station, but both are non-JR and not covered by the Japan Rail Pass. But don't worry. It costs only 150 yen from Koganecho Station to Yokohama Station. The fare of the municipal subway is 200-230 yen to the city center. It gets not only to Yokohama Station, but also the even more centrally located Kannai and Sakuragicho Stations.
You can also walk from your hotel to the city center in about 20-30 minutes.