First, I'd like to say that this quest SUCKS! I went through about 35 barge tickets before finally getting the lumber. Ironically enough I didn't get it while on the barge. I got it from a fished up "Fishtrap" (different from the "Flytrap"s that spawn normally) while waiting for the barge at Central Landing. I had only 4 fishing skill and used a Yew Fishing Rod and Lugworms to catch multiple of this mob. You can buy the rod and worms for a few hundred gil from Lusiane in South San d'Oria location F-8 (standing outside along the street west of AH, easy to find). I also tried the Willow Fishing Rod and Little Worms that NPC sells but those only got me Crayfish and a few Tricolored Carp.
I highly suggest using that rod/bait combo and fishing for monsters rather than riding the barge and waiting for random spawns. I got some type of monster or a rusty bucket every catch, no normal fish. You can fish while on the barge but I but I only got bites with that gear while in the Central Canal. In contrast, I was getting near constant bites when I finally tried fishing off the Central Landing dock. Oh, and /THF can't hurt if you are high enough level, I had it subbed when I got the drop (but I also had it subbed for about 2/3rds of my barge rides). Did this as 68MNK/20THF.
