Hiking Mount Fuji on your own is actually pretty simple and you have lots of options. Unfortunately it is not foolproof and there are some things you should know. Sadly,I did not make it to the top and will need to try again next year.
**Update** After not going with my family as a child and turning around in 2018, I did finally make it to the top August 2019! It was well worth the experience. **
If hiking up and back in one day, to make sure you have enough time to hike in the day light the most important tip I can give you is to start early. This means leaving the 5th station around 5 am. This is where I ran into trouble. I was very proud that we drove to Fuji and were parking about 5:10 am. However, I didn’t understand I was at the base of the mountain, I still needed to take a shuttle bus that only leaves every half hour. For us the 5:30 was already full which meant we caught the 6 am, and it was still a 45 minute ride to the 5th station. As you can see, this means we didn’t start walking until after 7 am.
There are a couple of other options to give yourself more time. You can start in the afternoon and reserve a spot at one of the many lodges around the 8th station. This means you can start early the next day and if the weather is good, see the sun rise. Or my plan for next year is to also go up the evening prior but spend the night at one of the lodges at the 5th station. This will allow me to acclimate and get an early start without dealing with driving and the bus. Or, as I should have done this year, arrive at the base of the mountain early (I would recommend 4 am) so you have time to catch the first bus and start walking with the sun.
Regardless of your plan, to get to Mount Fuji you can take a tour bus, drive yourself or take the train. For the train, your destination is Kawaguchikoguchi 5th station or Gotenba Station. For driving, you want to follow directions to 5597-84 Kamiyoshida Fujiyoshidashi Yamanashi. It is 1000 yen to park and 1000 yen for the shuttle bus from the parking lot to the 5th station.
This year I was afraid I would run out of daylight on the trip down and I turned around before reaching the summit. It was the right decision and I still had a wonderful day hiking. But next year, I will have a better plan. Fuji is only open to hiking mid-July through early September but is well worth it.