Don't know if this helps as it was a LONG time ago (1999 I think).
The taxi cost me almost 300,000 lit (that's about 150 Euros), it was about an hour's drive, Maybe less. The driver charged me for both ways.
Civitavecchia port is big and it can be quite a way to the exit depending on where your ship docks, our ship used to be about 10 minutes walk from the gate.
Taxis will come to the piers but I think you will have to order one, they're not there just waiting.
I went to the little cafe in the port and got the barman to call me a cab while I had a coffee, because my Italian's not fantastic....