I was considering taking a train from the Roma Ostiense station that leaves shortly after midnight and arrives in Pisa around 3:30 AM.

I want to see the Leaning Tower and get there right when it opens, which is 9:30 in the winter, along with exploring the town a bit before heading to Florence.

How early do shops and things open in Pisa? Is there a 24 hour cafe or coffee shop I could hang out in until possibly sunrise where I could walk around and explore the town?

This train is 20 Euros less than the other train if there was anywhere I could spend some time that early in the morning.

Thanks in advance for everyone's help.

Pisa is a city with a lot of students, and cafés, bars.
You might find something open - I didn't stay that late but last time I was there at 1 am, it was full of people.
I suppose some will still be there at 3 30 am.

Places that open early are usually unexciting places around main train stations but perhaps not that early.
Are you a student? Do you sleep? Are you trying to save hotel/hostel cost?

Are you looking at Intercity Notte 19€?

I don't know which day you are traveling, but I pulled a day out of hat and see something like this:

November 20, Termini-Pisa Centrale, 6:15-8:23, supereconomy 9€, Frecciabianca 35758

I am not a student, if I could save the hotel cost for tha night, that's great but not necessarily a must. I don't sleep that well generally, but thought I could possibly catch a bit of sleep on the train heading there?

The date I was looking at was November 5, departing Roma Ostiense station at 12:03 AM and arriving at Pisa Centrale at 3:23 AM, Intercity for €9.

I didn't know if there was anything that would be open that early and if there would be anything nice to see or look at if I walked around during sunrise.

There is an earlier train I see, also for €9, departing from the Rome Ostiense station at 7:41 PM and arriving at Pisa Centrale and 10:47 PM, so I suppose I could take that one, stay in Pisa, and then get up first thing in the morning for the tower.

Early november is quite not a time to party outside, not sure you'd have a lot of people in the streets.
Last I was in Pisa in nov, I just stayed in the hotel.

I wasn't so much looking for a party, just thinking I could possibly enjoy some of the sights before it got too crowded, but if there's not any cafes or anything open, not necessarily trying to wander around aimlessly in the dark either haha.

If I catch the late train and stay overnight, if I'm at the Tower when it opens, can I buy the tickets to climb the tower on the spot or should I still get them in advance?

Tickets are sold in timeslots as there are limited numbers allowed up the stairs. You won't be first in if all the numbers for the first timeslot are sold in advance.