Hey there Melissa. I'm going to do my best to answer your question, assuming it's not too late by now. The best way (without your own car) is going to be by bus. I did this trip in the opposite direction so I'll write this as if I was going the way you are looking to go. I had to take three (yep..) separate buses on this trip. The bus line I took started from Gora station in Hakone and ended at the Gotemba Premium Outlets. From there you would have to change to a bus that would take you to Gotemba Train Station. And from there you would need to get on a bus that would take you to Mt. Fuji Station in Fujiyoshida. Once arriving at Mt. Fuji Station, the best way forward will be to take the train to Shimoyoshida Station. Then you can hike up to the park/Shrine/overlook area. At the time I was using the Hakone Free Pass, so all these bus trips and train travel was "free", as part of purchasing the pass.

There is a road that leads up to the pagoda but they are closed sometimes. I am unable to tell you the opening/closing times though.

You can catch a Fujikyu line train from Kawaguchiko station to Shimoyoshida station. Trains run approx every 20 mins or so. First trains around 5:15am!!!

The best way to look up public transporration schedules is through hyperdia. All stations have coin lockers so eaay peasy!!