How do I apply for the residence permit within 8 days?

Short answer: Within eight working days of entering Italy, send the postal kit to the Questura of your city, through a post office or your university's permit desk (MUR circular). It costs €116.46 for a permit of up to one year.

Buy a €16 revenue stamp (marca da bollo) at a tobacconist.
Pick up the free kit at a post office with a "Sportello Amico" counter.
Fill in Modulo 1 in black pen and capital letters, with the stamp on the first page.
Put copies, not originals, in the envelope: all passport pages with data, visa and entry stamp; pre enrolment or enrolment certificate; proof of financial means; health insurance; accommodation proof; codice fiscale if you have one.
Pay €70.46 for the permit card plus €30 postal fee at the counter.
Keep the receipt (ricevuta) with your user ID and password, and the letter with your Questura appointment.

Sources: University of Pisa, University of Florence. Patronati (welfare offices) help with the kit for free (MUR circular).

This page explains official rules in plain language. I am not an immigration adviser and I do not give visa advice. For that, always use official government sources. Rules change: confirm with your university, the embassy and VFS before you act.