1) Go to your Steam Library (game list) and find Rome 2
2) Right click on Rome 2 and select 'Properties' (usually the last option in the right click menu)
3) Go to the tab called 'Local Files'
4) Click "Verify integrity of Game Cache"
5) Wait for Steam to scan your files and it should download the missing steam_api.dll file.
6) When Steam has finished, run the game