This idea already exists in the United States, where University and High School students get a 50% off discount on Spotify Premium and pay $4.99/month instead.
Now, Spotify has only launched in Japan, but they do not yet have a plan for the student discount. They did match the US price to make Spotify Premium normal 999 yen, but the 499 yen student price tier (does not yet) exist.
If we look at say a reason from a prespective other than a student, it is for example, good for society. This is because (like shown in other countries with similar music streaming services), when there is a convenient solution to listen to all music, illegal downloads (which are against the law in Japan) will decrease, and Spotify will even pay royalities to record labels and artists! (It is worth noting that students and the younger adults are probably the most responsibile for music piracy throughout the world.)
Further, the verification should be as easy as submitting an academic @...ac.jp e-mail address, perhaps every 12 months as in the US process for verification. I would not expect such an e-mail verification to be difficult. (However, if it proves so, I will gladly apply to be an intern at this company.)
Finally, I will make a seperate proposal for this, but I think in Japan, it is quite beneficial to offer an option to pay at the convenience store with the "Internet Purchase" option. This should increase your outreach and make people feel like it would be worth paying (no need for a credit card, etc) and should increase the company's customer base.
Updated on 2017-05-17
In case you hadn't heard @IceWobs this has been implemented! Let us know if you have any other ideas about Spotify in Japan, thanks!