Who found out that there were papers

The solicitor who owned the law firm knew the papers were there, because he'd gone there when Alice was alive and had been handling her legal affairs. He didn't want to go himself because, well, EEL MARSH HOUSE and WOMAN IN BLACK! He sent Arthur because he was also a solicitor, but an underling. Someone from the firm had to go, who was also a solicitor, not just a clerk.

Arthur's boss at the law firm had been to Eel Marsh House when Alice was alive, and wasn't keen on making the trip again. Whether he knew about the Woman in Black is debatable, but he presumably knew about the various deaths that had taken place at the house, and was visibly shaken when he mentioned Nathaniel's death. So he would have been keen to delegate.

He said that Jerome (the local solicitor in Crythin Gifford) was handling Alice's affairs on their behalf, but "he's not really been cooperating" (for obvious reasons). So they had to send one of their own out there to get the job done. Of course, travelling across the country to sort through some reclusive widow's dusty old paperwork in her tumbledown nightmare mansion was hardly a job for the up-and-comers at the firm, but a perfect opportunity for Arthur (the guy who wasn't pulling his weight) to redeem himself by doing the dirty job no one else wanted to do.

So basically the crap rolled downhill, and Arthur was at the bottom.