I take my board with me every time i go to Kagura or Gala (which is even more shinkansen friendly - the Gondola leaves from the shinkansen station)
Prior to this season i used a Burton Gig bag to carry my board and wear/boots etc. Havent had any problems with space; if you are quick you can load the board onto the overhead storage racks, otherwise just put it behind the back most seat in the carriage. This year i have a big booter backpack + board sleeve; i find I have a lot more mobility compared to the board bag, which always felt so huge.
Weekends in March will be crazy busy on that shinkansen though. And also when you get to Echigoyuzawa station, you need to catch a bus to Kagura. Catching a 6:30ish shinkansen from Tokyo gets you there before 8, but there have been times where the line for the bus is easily 100 people long. Somewhat painful for a bus that leaves every hour or so. If you are going with friends, you can just catch a taxi, but thats 3000yen each way.
Since you will only be there for one month i am guessing money wouldnt be much of a concern, but you can also look into the bus tours; the night buses leave Tokyo around 11pm and drop you off right at the snow resorts at 6am or so. Lets you get ready/snooze before hitting the first lines. They are also cheaper than Shinkansen (6-8000yen vs 13+k), though if you have trouble sleeping on a bus it is quite rough.
My friends also post their boards to and from the resorts; its only about 1500yen each way. I prefer to have my gear with me from the instant i get there, and not have to wait 2 days to dry out my board wear. And since i try to get out on the slopes every weekend from November to May, i prefer to save money where i can.