One point to clarify....
Are you using the train to Monterosso directly after arriving on an international flight? One important point to remember is that train reservations are specific to train, date and departure time. If your flight is late arriving and you miss the train you've booked, do NOT try to use the tickets on another train, as you'll be subject to hefty fines which will be collected on the spot.
OTOH, if you're connecting with that train from another location in Italy, buying advance tickets can save some money. I agree with the suggestion in the previous reply that it's better to buy from the Trenitalia website. I've also used Trainline Europe in the past and found that worked very well (although I believe they may have a small surcharge). Both Trenitalia and Trainline have smartphone apps, so you can buy tickets using your phone (cellular access required if no WiFi available).