Osorezan is located in the middle of the Shimokita peninsula: it is therefore a remote place from the Kantō region, not to mention from Kansai. A day trip from these two regions is thus unthinkable.
This is possible if you leave from the nearby Aomori city or, in a little more time, from Morioka, in Iwate Prefecture.
As suggestion by us, the Shimokita Peninsula, however, requires more than one night stay to be explored and enjoyed, as it is full of places to see, such as Shiriyazaki, Cape Ōma, Hokoke-ga-Ura and more!
To get to the Shimokita peninsula from Tōkyō, take the Tōhoku Shinkansen to Hachinohe Station. After getting off, hop on an Aoimori Electric Railway train (the latter is not covered by the all-country covering Japan Rail Pass, but it is by the East Japan Rail Pass) to Noheji Station. From there, continue on the JR East Ominato Line to Shimokita Station, where you will find a bus route to Osorezan from the Mutsu Bus Terminal.
As you can see, traveling by public transportation could be a bit tricky, so if you can manage to rent a car, maybe is better (and it is definitely necessary if you want to see the whole peninsula in reasonable times).
If you are traveling by plane, fly from Haneda Airport to Misawa Airport and, once landed, continue by bus to Misawa Station. With the Aoimori Electric Railway you arrive at Noheji station. From here, proceed as described above.
Pay attention to the fact that * the Osorezan can only be visited from May to October *, due to the climatic conditions of the colder months.