I will be arriving at the Pisa airport around 3:35PM on Sunday, June 30 on a Ryan Air flight. I want to get to Monterosso that day. Because it is a Sunday, it seems that there are not many trains. I might not be looking at all of the possibilities. What train will get me to Monterosso that day?
Don't worry that is easy I did that route on a day trip.
Just search Pisa Centrale to Monterosso. There is maybe a change of trains in La Spezia.
It might be that the schedule for June 30 has not been posted yet. It may be as late as June 8 before the complete schedules are posted. I checked for the first Sunday in May and there are oodles of trains after 16:00.
I'm guessing that you're not seeing much because the train schedules change earlier in June. Suggest you try the Trenitalia site, looking at the schedule for just a few days from now (the schedule doesn't change much). Be sure to set not only the date, but also the earliest time you might be able to catch the train.
Several trains leaving after 4:00. Take the shuttle from the airport to Pisa Centrale. At that time of the day it won't be very crowded on the trains.