There are several routes up the mountain, but the most common is from Adatara Kougen, where a ski resort is located. In season, there is a gondola which will take you from the bottom of the resort (950 meters) up to an elevation of 1350 meters, shaving off about an hour of climbing and half of your total ascent. Adatara Kougen is accessed using route 459 which intersects the expressway at Nihonmatsu IC. Typing "あだたらこ" into my navi brought up the right option. There is plenty of parking at Adatara Kougen. I do not believe there is public transportation that far, but a bus does seem to run to the Adatara Kougen Fukishikyu Hotel (あだたら高原富士急ホテル) marked on the map. That would add a couple extra hours of walking each way, though, so it is not really feasible. Opening day of climbing season for 2014 was May 18th. I climbed on May 31st, and there was one trivial patch of snow to traverse.
Once at Adatara Kougen, you can either take the gondola (1200 yen round trip, 950 one-way) or just hike up the ski resort. Hiking takes about an hour to reach the top of the gondola. Take any ski run you wish or just follow others. They all connect at a top, but the easily-found trail leading from the last run to the gondola station is on the top left corner of the ski run area. A map of the area is shown below. There is a trail through the forest on the right (north) side of the ski runs that I did not take, as all the other hikers were going up the runs, which I have crudely marked in green. Anyway, it is easy to figure out - just walk up the hill and aim for the top left corner, or follow that trail in the woods. Note that north is more or less right on this map.
From the area near the gondola station, the trail is easy to figure out. Just follow the signs for 安達太良山頂, or alternatively, follow the crowds as you are all going to the same place. The hike from the gondola station to the top takes about 75 minutes.
From there, you are golden. I hope you left early, because this is a great ridge hike. First, climb up the to the very summit and bag your hyakumeizan, which is right in front of you when you get to the top.
Once you have made it to the top, climb off the rock pile and start walking around Numanodaira crater, the site of the explosion. I spent about four hours doing a horseshoe around it, heading over to Tetsuyama and the shelter hut, and Funamyojin. To give you a taste of what you will see, here is one of my photos of the crater.
Numanodaira crater, with Bandai-san (another hyakumeizan) in the distance
When you are done wandering around the rim, just take the same route back down (with or without the gondola), which should take you about 1:50 without and 1:05 with a ride. Alternatively, you can hike down towards Mt Kago and then take the more northern trail back to your starting point (Okudake trailhead, 奥岳登山口) which I would guess would take about the same amount of time. Lots of people seemed to descend on that trail after having ridden up on the gondola.
Anyway you cut it, this a great hike, with a high payoff for not so much slogging. Keep sure you have plenty of time to wander around the rim!