There are many ways to get between Ulsan and Busan. One of the more budget friendly ways to get between the two cities is to take an intercity bus. The bus leaves from Nopo Bus Terminal which is on subway line 1 (the orange line). The buses are outside and don't depart from the same location as the other buses heading to cities further away. The two buses that I know you can take to Ulsan are 1127 & 1137. The one way
fare if paying with cash is 2,000 won and if paying with the Busan transportation card is 1,800 won. The ride takes about 1.5 hours.
You can also get to Ulsan via the trains and express buses. The one way fares for the train and express bus ranges between 6,000-7,000 won.