Hi there!
The Fujikyu services to and from Kawaguchiko use platforms 1 & 2 and the JR services use the remaining platforms 3-5. As you'll be arriving from Tokyo, I expect your train to pull into either platform 3 or 4 depending on the train you get.
As platform 1 & 2 and 3,4&5 sre on separate island platforms, you'll need to change platforms to transfer trains. Otsuki is not a big station and as part of this transfer process in the past I have purchased tickets for the Fujikyu component of the line when I transfer however you may be able to purchase the Fujikyu component by itself at the JR ticket counter in Shinjuku before you depart - it's worth asking as I think the process of changing train and purchasing ticket at Otsuki will take around 10 minutes - longer than the transfer time you specified. If you can't purchase before you get to Otsuki you don't need to worry too much as there are trains between otsuki and kawaguchiko every 25 minutes.
By the way, JR operate a direct service to Kawaguchiko from Shinjuku called the 'Holiday Rapid Kawaguchiko' which only operate at the weekends and public holidays. These trains only run in each direction twice per day (9:35, 9:59 from Shinjuku, 16:00 & 18:24 on the return). While you would still need to pay the Fujikyu line fare (either on the train, or before you depart at the JR ticket office), it could save you time and energy if you use it as there is no transfer/ticket purchase at Otsuki. Here's the itinerary for the trip from Shinjuku to Kawaguchiko using these JR trains.
Hope this helps!
