From the center of Rome, buy a metro ticket (1,50 euros) at newstands or metro stations and take the B line. Get off at PIRAMIDE.
STAY INSIDE THE STATION and look for the signs 'TRENI TO OSTIA' or 'ROMA-LIDO', as shown below.
The ticket that you bought before is valid 100 minutes and will include the ride to Ostia Antica. Trains leave every 10/15 minutes and an electronic board will show which platform (binario) to reach. By the way, this is the first italian DOUBLE-TRACK railway, inaugurated by Mussolini in 1924 to connect the Capital to the beach.
Once on board, enjoy your 25 min ride to Ostia Antica Station and see romans coming back home from work or on the way to the beach. Notice that the train goes from Porta S.Paolo (your starting point) to Cristoforo Colombo (the beach stop).
Outside of the Ostia Antica station you will see a blue pedestrian bridge that you need to take to cross over the road and then follow the signs to the entrance of the excavations (10 min walk) where you can buy your ticket (10 euros) and meet me right in front of the ticket office. The site opens at 8.30 and is closed on Mondays.