Piraeus Port

You don't have to leave the port to get into Central Athens. There are buses right at the port as well as the Green Line Train at the Piraeus Metro Station, which is also right at the port near the Blue Star Ferries for the Cyclades islands. The train will take you to the Monastiraki Metro Station, which is right on the edge of the Plaka. It will all become clear once you get there and see the layout. If you're arriving on a cruise ship you will be about 20 minutes' walk along the waterfront to the Metro Station. I'm not sure, but there may be a shuttle bus that will take you from the cruise dock to the Metro Station.

Lee is right on the money with everything. Monastiraki station is a reasonable walk to the Acropolis, and you can do a very short detour to the Roman Agora as well, which I particularly enjoyed. To make the most of your time, consider taking the metro into town, seeing the sites, and then taking a taxi back to the ship. Otherwise, after you see the Acropolis, go down the other side, see the new museum at the bottom & across the street. Then get the red line and change to the green at Omonia stop. It is pretty simple to do.