Shin-Yokohama is quite a small station, with only the Shinkansen and Yokohama line, as well as an underground station. However, the route to the Yokohama line is a bit involved as it is in a separate part of the station and requires 3 sets of escalators/stairs, which get crowded at peak times, so you might struggle, especially if you have luggage.

I suspect that you're doing your train search on Hyperdia.com using Kyoto as your Start Point and Kawaguchiko as your Destination in arriving at your train schedules and the short transfer times. The transfer time at Hachioji Station is also very short. Under "Time Table" for all trains, click on the picture of the clock & book and you will see that the next trains are only 8 to 30 minutes later. So, catching the next train is a better option.

For your train search, do not use Kyoto with final destination Kawaguchiko. Instead, do your 1st search from Kyoto to Shin-Yokohama, your 2nd search from Shin-Yokohama to Hachioji, your 3rd search from Hachioji to Takao, etc. Then, you can catch the later trains and give yourself more time for each transfer.

Do your train scheduling as I suggested and you'll be fine.

In Hachioji Station you just move to a neghbouning platform. Most of the non-express trains going westbound are destined to Takao. Some of the Ltd. Express train do not stop at Otsuki.

It's better if you can get Fujikyu tickets in advance, via a major travel agent's branch in Kyoto, for example.

Have a nice trip!

I'll do the search as Harold suggested, thanks so much for that! And I'm relieved to hear that at Hachioji the platforms are so close. Phew!

I'll be travelling on New Year's Day actually, and I'm very anxious about it after reading the forums & the site! I'm hoping to get to Kawaguchiko before 5pm.

Thanks again =)