There are buses from the Main station to Fulong. The ride along the coast once past Keelung is really nice. But the easiest way is to go by Train from Taipei Main Station. Take the MRT to the Main Station. Then buy tickets at the TRA ticket station on the main floor.
Be aware that Taipei Main Station has the MRT subway, the TRA regular trains and the HSR high speed rail.
You want to get tickets for the TRA trains heading to the East Coast of Taiwan. They have different ticket stations for WEst Taiwan and East Taiwan.
Most any trains stop in Fulung. Take the Taroko Express OR the new Puyuma Express (newer version of the Taroko express).
Will be less then an hour with these trains. Up to 90 mins on slower trains. The train ride is nice too. Good scenery and lots of tunnels.
It may be easier to go there by train and then take the bus back. Just board any bus that said KEELUNG (and transfer from there to Taipei via many local trains or buses) or TAIPEI. The TAIPEI buses will likely be heading towards ChongShiao E. Rd and/or Taipei Main station.
There are buses from Keelung to Fulung as well.
oh, someone already posted bout the train