Its certainly possible to travel on some container ships, there are even books written about how to do it. However the two major limitations are there aren't that many ships that are prepared to do it these days, in part due to regulations such as over a small number of passengers (say 10 for example, not sure on the exact figure), they have to have a doctor on board, other facilities, etc.
The other far more significant factor is its now a somewhat exotic way to travel and hence its priced accordingly. The days of hopping on tramp steamers around the world have gone. Additionally it will be a further challenge to link up a ship that can take your vehicle in a container with one that is prepared to take passengers.
Hope it works out and you find something but suspect it might not be possible for a cheap price.