Ulsan Airport and KTX Ulsan Station are separate transport hubs. If your next step is a train, first confirm whether your ticket uses Taehwagang Station or KTX Ulsan Station rather than asking simply for 'Ulsan Station.'
Korea Airports Corporation currently says the city/intercity bus stop is about 150 meters straight ahead after exiting the passenger terminal. The taxi stand is across the crosswalk from the arrivals gate. Check the current bus route before leaving the terminal, because the best bus depends on your exact district or rail station.
The airport's current taxi estimates illustrate the geography: about 20 minutes and KRW 9,000 to Taehwagang Station, about 30 minutes and KRW 11,000 to Ulsan City Hall, and about 40 minutes and KRW 24,000 to KTX Ulsan Station. These are estimates and can change with traffic.
If you are connecting to high-speed rail, budget the airport-to-KTX transfer as a real journey rather than a terminal change. If your destination is downtown or the east-side industrial/coastal area, KTX Ulsan Station may be the wrong intermediate hub entirely. Use the airport's current bus information or a local route planner for the exact stop and recheck the return trip before an early flight.