I have no idea about taxi prices ... However, there is direct bus to South Stack about 350metres away.
Bus travel is much better and less frantic, I find.
How to get to the bus? The bus stop is at Summer Hill in Holyhead town and is easy to get to by foot from the station.
Follow the rails to their very end inside the station. As the old building transitions into the ferry terminal, you'll see a sliding door on your left. Go through it and travel across the Celtic Gateway Bridge into town.
Immediately into town turn right and follow the pedestrian walkway and street to its very end. The bus stop is up on your left.
The bus you require is Bus 22 and it stops at the RSPB Information Centre.
Outgoing bus at 10:45, 14:30 and 17:00. Return times from South Stack are 11:00, 14:45 and 17:15.
Hope that's useful. South Stack is a lovely place to visit for scenery, the lighthouse and the birds.