Rather than to walk to a train station and travel all the way backwards to (the very busy) Yokohama Station, take a quick taxi ride from your ship to Ishikawacho Station. Four ordinary-size adult passengers can legally squeeze into one taxi. You can also check to see if you can catch the semi-touristic Akaikutsu (red shoes) Bus from your port to either Sakuragicho Station or Yokohama Station.
http://www.city.yokohama.lg.jp/koutuu/kankou/sanrosen/pdf/burari-reafl...From any of the above 3 stations, you can take the JR Keihin-Tohoku Line and change at Ofuna Station for JR Yokosuka Line (from Yokohama Station, you can also take the Yokosuka Line, and then you won't have to transfer any more). From Ishikawacho Station to Kita-Kamakura Station, it takes about 30 minutes including transfer. And then Kamakura Station is 1 station further from Kita-Kamakura which takes 3 min by the same Yokosuka Line train, unless you decide to take a stroll there like many tourists do.
At Kamakura Station, you can catch the cute Enoden train. Try to queue for the next next train so that you can sit/stand on the very front or rear of the train which provide great views of the cozy little track. You might want to consider purchasing one of their passes depending on how you'd like to enjoy your day.
https://www.enoden.co.jp/en/If you have your mind set on Enoden, you may want to do that first. There are lots of tiny temples and tiny towns along the track that you may want to take a better look at. Along the track, the island of Enoshima is also fun, except that it will be packed with young people on weekends and warm weekdays. Then, coming back, you may want to do some extra walking around the vast Kamakura Station area, or the more peaceful Kita-Kamakura area.
Back at the harbor, Ishikawacho Station is the closest JR station to China Town, and the harbor area also has a variety of nice places to dine on a variety of budgets. Sakuragicho Station is where domestic tourists tend to go for modern sites on the harbor side, or where residents prefer to explore on the other side for tiny bars.
Keep in mind, however, the weather. Hard rain is not suitable for walking, so in that case you may want to restrict yourselves to train/bus windows and museums or even bus tours if any.
From Tokyo, there are no particular express trains for Kamakura that would skip Yokohama, so technically speaking, your port is closer to the area. But if you're planning to go to Kyoto, you may find that Kamakura is similar. They're both old capitals except that Kamakura is the one with the sea.
Bon voyage!
