Even though Arthur is shaking in his boots throughout his whole miserable trip to Crythin Gifford and Eel Marsh House, he's still on a quest to find the truth. The Woman in Black gives us plenty of mysterious loose ends, and Arthur is the slightly terrified man for the job. He has to both literally and metaphorically explore Eel Marsh House and its former inhabitants to find out the who, what, where, and why of Crythin Gifford's haunted past.
Even though Arthur could have come up with an excuse to leave Eel Marsh House and Crythin Gifford, he stays because he wants to solve the mystery of the old house.
Arthur is as much the victim of a horror movie as he is a protagonist of a mystery; he spends as much time running through the house in terror as he does sorting through papers to find the truth.