I did this in 2012 from Australia to South Korea - theres a bunch of specialized travel agencies you can book through. From memory it took about 10 days.

The cost was about $100 / day, but this was almost 10 years ago, so I guess it went up by now. This isn't really competitive as travel, but if you consider the experience of being on a container ship a holiday in itself, the cost was quite reasonable IMO.

There was one other passenger on board, and we could go anywhere on the ship without an escort except the engine room. I really enjoyed it, but its not for everyone as there is no communication / internet access available so you have to be the sort of person who can entertain yourself.

Also wouldn't go if you get seasick easily. Unlike say cruise ships, the crew isn't too worried about passenger comfort, and wil just plow straight through storms.