It depends on whether you count a bucket as a tool or not. Regardless, if you find one as loot, you can use it.
As for a farm having to be inside the wall or not, the wording suggests the castle wall only needs to surround the buildings and a farm isn't seen as a building. The requirements for the wood age flat out state the wall only has to contain the keep and the spawnpoint, and not the farm.
Remember though, this challenge is for fun. If you can't feasibly put a farm inside your walls without the use of a bucket, I don't think people will mind if you set it up at a more remote location.
I'd recommend not using the bed at all. Night time is a great time for grinding EXP, mob drops, and just being productive. I do not see the point of sleeping a night that you can spend placing blocks, swinging your blade, or constructing redstone mechanisms.
Try using CastleBuildingChallange as your seed. It put me on an isolated jungle island with only sheep and cats for animals. Plenty of wood around but not much else. Only had apples to eat till the farm got going.