Barbara,
That trip is very easy! There are several direct (no changes) trains from La Spezia Centrale to Rome every day, so you don't have to limit yourself to the departure you mentioned. However that will be one of the quickest solutions, so these are the times for the entire route....
- 07:35 depart Monterosso (via Regionale - DON'T forget to validate)
- 07:50 arrive La Spezia
- 08:18 depart La Spezia (via FrecciaBianca)
- 12:03 arrive Roma Termini
As you can see, you can absolutely get there "in time" from Monterosso using one of the local trains, as it's only a 15 minute trip to La Spezia Centrale. On the above trip, you can save a few dollars if you buy an advance ticket for the Freccia train (I usually buy from www.captaintrain.com as the website is very user friendly). For the first segment on the Regionale, just buy your ticket in Monterosso a day or two before your travelling. The ticket doesn't become "usable" until you validate it.