Can you store luggage there for a few hours
I'm sure consensus in this Travel Forum has been "no." You can store luggage at Pisa Centrale, further from the sites, but with a shuttle bus. Italian stations don't have lockers, so those stations that offer the service have to staff it.
There is only a baggage check desk at Pisa Centrale Station (not at Pisa San Rossore station).
Look for 'deposito bagagli', it's next to the police station at the far end of platform 1. (€4/bag for 12 hours, open daily 6am-9pm)
If you plan to climb the Tower, you might want to book your tickets in advance, otherwise you may have to wait: http://www.opapisa.it
Enjoy your trip!