Take an expressway bus bound for Kawaguchiko or Fuji-Q Highland from Shinjuku Expressway Bus Terminal, and get off at Chuo EXPWY Shimo Yoshida Bus stop. Price one way: 1,700JPY.
https://highway-buses.jp/course/kawaguchiko.php
Approx. 20 minute walk will bring you to the Arakurayama Sengen Park.
(When you get off the bus, please walk to the left, then you will find a direction to the destination.)
The service which is bound for or leaving from Shibuya does not stop at Chuodo-shita Yoshida Bus Stop.
Unfortunately you cannot switch to the next bus with the same ticket.
There is a train from Shimoyoshida station(close to Arakurayama sengen park, shiogama shrine, omuro sengen) that will take you to FujiQ Highland station which cost around 220 yen.
Otherwise you could use a taxi service. For this short distance it should not be very expensive.
To make a taxi arrangement in advance online, see:
http://www.fujiq-taxi.jp/
Unfortunately the page is only in Japanese. It might be better to make a phone call to 0120-818-229 when you need to charter a taxi.
Hope this can help you make a plan.
Have a nice trip in Japan!