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Would this slight mismatch be any problem?
It seems unlikely. The actual start date of the program is later than the date from which your university has indicated in the certificate of no objection.
Can my visa get rejected because I intend to reach on 8 June, while the university letter mentions 15 June? How can I explain them my intention?
They're more likely to give you a visa valid from 14 June than to reject it altogether. But if you present an itinerary arriving a week before, I suppose they're likely to find that reasonable and issue the visa accordingly.
Another question you didn't ask:
Will they grant the visa for 8 June even though my university's certificate of no objection mentions 12 June as the program start date?
If the other applicant's experience with the German consulate is any indication, this could be a problem, although I suspect it is not actually very likely.
Presumably if the university has no objection to your being in the Netherlands on June 12th, they have no objection to your being absent from their own campus for a day of travel on the 11th. You can also include a copy of your university's academic calendar or other evidence explaining why your presence is not required there between the 7th and the 11th.